Fully Coupled Pauli-Fierz Hamiltonians at Zero and
Positive Temperature
—————
Summer school on Non-equilibrium Statistical
Mechanics
—————
Montreal 2011
First Draft
Jacob Schach Møller
Department of Mathematics
Aarhus University
Denmark
July 24, 2011
Contents
1 Introduction
2 Construction of Operators
2.1 The Hamiltonian . . . .
2.2 The Standard Liouvillean
2.3 Jakšić-Pillet Gluing . . .
2.4 Multiple Reservoirs . . .
2.5 Open Problems I . . . .
2
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2
2
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3 Bound States
3.1 Number Bounds at Zero Temperature . . . . . . .
3.2 Number Bounds at Positive Temperature . . . . . .
3.3 Virial Theorem’s . . . . . . . . . . . . . . . . . .
3.4 A Review of Existence and Non-existence Results .
3.5 Open Problems II . . . . . . . . . . . . . . . . . .
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45
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4 Commutator Estimates
4.1 The Weak Coupling Regime . . .
4.2 Conjugate Operators . . . . . . .
4.3 Estimates at Zero Temperature . .
4.4 Estimates at Positive Temperature
4.5 Open Problems III . . . . . . . . .
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1 Introduction
The purpose of these notes is to give a fairly narrow but thorough introduction
to the spectral analysis of Hamiltonians and Liouvilleans describing finite dimensional small systems linearly coupled to a scalar massless field or reservoir. The
Hamiltonians describe the system at zero temperature, and the standard Liouvillean
implements unitarily the dynamics of the system at positive temperature.
We focus our attention on results valid at arbitrary coupling strength and whose
proofs are purely operator theoretic, i.e. for the standard Liouvillean, does not
make use of the underlying modular structure. For the Liouvillean this means that
important structure results that does not seem to have a purely operator theoretic
proof, like Jadczyk’s theorem, will only be mentioned in passing.
2 Construction of Operators
2.1 The Hamiltonian
As a small quantum system we take a finite dimensional Hilbert space H = Cν
with Hamiltonian K ∈ Mν (C), a self-adjoint ν × ν matrix K ∗ = K. In fact we
can, without loss of generality, choose K to be diagonal with its real eigenvalues
sitting on the diagonal.
The dispersion relation for the field is the massless relativistic relation k → |k|
considered as a multiplication operator on h = L2 (R3 ). This gives rise to the
second quantized free field energy Hph = dΓ(|k|), as a self-adjoint operator on
⊗s ℓ . We write |0i = (1, 0, 0, . . . ) for
the bosonic Fock-space F = Γ(h) = ⊕∞
ℓ=0 h
the vacuum state in F .
We define a class of admissible coupling operators/functions
G ∈ B(K; K ⊗ h) = L2 R3 ; Mν (C) .
That the two spaces above can be identified can be seen as follows: If G : R3 →
Mν (C) is square integrable one can define a bounded operator BG ∈ B(K; K ⊗ h)
by
(BG v)(k) = G(k)v,
where we identified K ⊗ h isometrically with L2 (R3 ; Cν ). Then
Z
2
2
kBG k = sup kBG vk = sup
|G(k)v|2 dk ≤ kGk2 .
|v|≤1
|v|≤1 R3
and the linear map G → BG is a contraction, but it is not an isometry. To see
that it is surjective with a bounded inverse, let B ∈ B(K; K ⊗ h) and define the
2
candidate for an inverse G by Gij (k) = h(Bei )(k), ej i, where e1 , . . . , eν is the
standard basis for Cν . Then
Z
ν Z
X
2
|Gij (k)| dk =
|(Bei )(k)|2 dk = kBei k2K⊗h ≤ kBk2 .
j=1
R3
R3
Hence
2
kGk =
Z
2
R3
kG(k)k dk ≤
X Z
1≤i,j≤ν
R3
|Gij (k)|2 dk ≤ νkBk2 .
From now on we will identify couplings G with elements of L2 (R3 ; Mν (C)), and
norms of couplings will be L2 -norms. We remark that the identification of coupling
operators as B(K)-valued functions above is particular to finite dimensional small
systems, cf. [16, Remark 5.1]. Let µ > 0 be arbitrary, but fixed. For the coupling
G we assume the existence of a constant C > 0 such that
(HGn)
∀k ∈ R3 , |k| ≤ 1,
∀k ∈ R3 , |k| ≥ 1,
and
|α| ≤ n :
and
|α| ≤ n :
3
k∂kα G(k)k ≤ C|k|n− 2 +µ−|α|
3
k∂kα G(k)k ≤ C|k|− 2 −µ .
The derivatives are distributional derivatives. We will make use of the condition
(HGn) on G with n = 0, 1.
The above conditions reflect that |k||α|−n ∂kα G is slightly better than square
integrable near zero, and the ∂kα G’s are slightly better than square integrable at
infinity. For our commutator estimates in Sect. 4 it will not suffice to demand just
square integrability. We remark that there is nothing special about three dimensions
or the dispersion |k|. For some results we could deal with infinite dimensional
small system K, and more singular G’s. The above special case however captures
the essentials, and permits us to formulate simple - yet pertinent - conditions that
can be used for all our results at zero temperature.
We now define the free and coupled Hamiltonians as
H0 = K ⊗ 1lF + 1lK ⊗ Hph
H = H0 + φ(G),
where
1
φ(G) = √
2
Z
R3
{G(k)∗ a(k) + G(k)a∗ (k)} dk.
We remark that H0 is self-adjoint on D(H0 ) = D(1lK ⊗ Hph ) and that
C = K ⊗ Γfin (C0∞ (R3 ))
3
(2.1)
is a core for H0 . Note the form bound on C
±φ(G) ≤ σN + 2σ −1 kGk2
±φ(G) ≤ σHph + 2σ −1 kG/
p
(2.2)
|k|k2
(2.3)
valid for all σ > 0 (and G). By (2.3) and Kato-Rellich’s theorem, [48, Thm. X.12],
H is essentially self-adjoint on C and D(H) = D(H0 ). In particular, the domain of
H does not depend on G. The notation Γfin (V ), with V ⊂ h a subspace, denotes
the algebraic direct sum of V ⊗s n , with tensor products of subspaces of Hilbert
spaces always being algebraic, whereas tensor products of Hilbert spaces always
denote Hilbert space tensor products, i.e. completion of algebraic tensor products.
The bound (2.3) in particular implies that H is bounded from below. We furthermoreRobserve that if we equip the space of G’s satisfying (HG0) with the norm
kGk20 = R3 (1 + |k|−1 )kG(k)k2 dk, then the resolvent map (z, G) → (H − z)−1
is norm continuous. Here Imz 6= 0. We define
Σ = inf σ(H) > −∞.
(2.4)
The spectrum of H is in fact a half-line starting at Σ as we now prove, using an
argument from [23]. Before we provide the proof we establish a version of the so
called pull through formula
√
Proposition 2.1. Suppose (HG0). For any z ∈ C\[Σ, ∞) and ψ ∈ D( N ) we
have as an L2 (R3 ; H)-identity
1
a(k)(H − z)−1 ψ = (H + |k| − z)−1 a(k)ψ − √ (H + |k| − z)−1 (G(k) ⊗ 1lF )ψ.
2
Remark 2.2. It is an immediate consequence of [43, Prop. II.1] that H is √
of class
1 (N ), implying in particular that D(N ), and hence by interpolation D( N ), is
CMo
preserved by resolvents of H. Hence both sides of the pull through formula defines
elements of L2 (R3 ; H). The commutator is [H, N ]◦ = iφ(iG). We note that it also
holds true that N is of class C 1 (H), something we will not make any use of.
Proof. Let ψ̃ ∈ C and compute
1
a(k)(H − z)ψ̃ = (H + |k| − z)a(k)ψ̃ + √ (G(k) ⊗ 1lF )ψ̃
2
as an L2 (R3 ; H)-identity, where the only possibly irregular contribution is G near
zero. Since z − |k| ∈ ρ(H) - the resolvent set for H - we obtain the L2 (R3 ; H)identity
1
a(k)ψ̃ = (H + |k| − z)−1 a(k)(H − z)ψ̃ − √ (H + |k| − z)−1 (G(k) ⊗ 1lF )ψ̃.
2
4
Let h ∈ L2 (R3 ) and ϕ ∈ C. Then
∗
a (h)ϕ, ψ̃ = ϕ̃, (H − z)ψ̃ −
where
ϕ̃ =
Z
R3
Z
R3
h(k) √ ϕ, (H + |k| − z)−1(G(k) ⊗ 1lF )ψ̃ dk,
2
h(k)a∗ (k)(H + |k| − z)−1 ϕ dk ∈ H.
From this expression, and H being essentially self-adjoint on C, we observe that
the above√identity remains true for ψ̃ ∈ D(H). Inserting ψ̃ = (H − z)−1 ψ, where
ψ ∈ D( N ) yields the proposition. Here we used that L2 (R3 ) ⊗ C (algebraic
tensor product) is dense in L2 (R3 ; H).
For stronger versions of the pull through formula see [10, 24]. We are now
ready to show that the spectrum is a half-axis. The argument goes back to [23], cf.
also [10].
Theorem 2.3. Suppose (HG0). Then σ(H) = [Σ, ∞).
Proof. It suffices to show that σ((H − Σ + 1)−1 ) ⊃ (0, 1]. To see this, let λ > 0,
ǫ > 0 and choose ψ̃ ∈ 1l[H ≤ Σ + ǫ/2]H to be normalized. Since C is dense in
D(H), we can pick a ψ ∈ C such that k(H − Σ)(ψ̃ − ψ)k ≤ ǫ/2 and hence we
must have k(H − Σ)ψk ≤ ǫ.
Choose a function h ∈ C0∞ (R) real-valued with khk = 1 and supp h ⊂
3/2
∗
[−1, 1]. Put
√ hn (k) = n h(n(|k| − λ)). Form ψn = a (hn )ψ and compute
for ϕ ∈ D( N ) using the pull through formula Proposition 2.1
ϕ, (H − Σ + 1)−1 − (λ + 1)−1 ψn
Z
=
hn (k) a(k) (H − Σ + 1)−1 − (λ + 1)−1 ϕ, ψ dk
3
ZR
=
hn (k) (H + |k| − Σ + 1)−1 − (λ + 1)−1 a(k)ϕ, ψ dk
R3
Z
hn (k)
√ G(k)dk ⊗ 1lF ϕ, (H + |k| − Σ + 1)−1 ψ .
+
2
R3
Since hn goes to zero weakly in L2 (R3 ), the last term is o(1)kϕkkψk in the limit
of large n. To deal with the first term on the right-hand side we estimate using the
5
support properties of hn and the choice of ψ:
hn (k) a(k) (H + |k| − Σ + 1)−1 − (λ + 1)−1 ϕ, ψ p
|hn (k)| (Hph + 2)−1 |k|a(k)ϕ
≤ p
|k|
× (Hph + 2)(H + |k| − Σ + 1)−1 (H − Σ) + (|k| − λ) ψ p
1 |hn (k)| |k|a(k)(Hph + 1)−1 ϕ.
p
≤C ǫ+
n
|k|
p
Noting that khn / |k|k ≤ (λ − 1/n)−1/2 khn k = (λ − 1/n)−1/2 we conclude
from Cauchy-Schwartz that
ϕ, (H − Σ + 1)−1 − (λ + 1)−1 ψn ≤ C(ǫ + o(1))kϕk,
where o(1) refers to the large n limit. It now remains to prove that ka∗ (hn )ψk is
bounded away from zero. But this follows from the computation
ka∗ (hn )ψk2 = kψk2 + ka(hn )ψk2 .
Recall that
√ when hn goes to zero weakly, we have a(hn )ψ → 0 in norm, whenever
ψ ∈ D( N ).
We end this subsection introducing some extra structure that will be used to
construct the standard Liouvillean in the next subsection.
We will need a conjugate linear involution operator C on H defined as follows. It is a tensor product of two conjugate linear involutions, one on K and one
on F . On K we simply take coordinate wise complex conjugation (cv)j = v̄j ,
and on F we take second quantized complex conjugation Γ(c), acting on an nparticle state by complex conjugation, or equivalently described by the intertwining Γ(c)a# (g)Γ(c) = a# (ḡ). In conclusion C = c ⊗ Γ(c). Note that hCψ, ϕi =
hCϕ, ψi.
With this choice of conjugation we can define H c = CHC = H0 + φ(G).
Note that H0c = CH0 C = H0 . Clearly, the spectrum, pure point spectrum and absolutely/singular continuous spectrum of the two operators coincide. Eigenvectors
are related by ψ c = Cψ, where Hψ = λψ and H c ψ c = λψ c . Finally we observe
that the spectral resolutions E and E c of the operators H and H c are related by
Eψc = ECψ .
We will use the notation N for the number operator dΓ(1lh ) as an operator
on F , and we recycle the same notation on H instead of the more cumbersome
1lK ⊗ N .
6
2.2 The Standard Liouvillean
The Liouvillean, at inverse temperature β > 0, is a self-adjoint operator on the
doubled Hilbert space HL := H ⊗ H. The zero temperature Liouvillean, corresponding to β = ∞, is given by
L∞ = H ⊗ 1lH − 1lH ⊗ H c ,
which is essentially self-adjoint on algebraic tensor products D ⊗D, where D ⊂ H
is a core for H. See [47, Thm. VIII.33]. As a choice of core we take
C L = C ⊗ C,
(2.5)
where C was defined in (2.1). Observe that L∞ is unbounded from below and
indeed σ(L∞ ) = R.
We furthermore write
L0 = H0 ⊗ 1lH − 1lH ⊗ H0
for the uncoupled Liouvillean. Recall that H0c = H0 . With this notation, at least
formally, the zero temperature (β = ∞) Liouvillean can be written as the operator
sum L∞ = L0 + φ(G) ⊗ 1lH − 1lH ⊗ φ(G).
We will need stronger conditions than (HGn) on the coupling G when dealing
with the Liouvillean. Let n ∈ N0 . We assume that G admits n distributional
derivatives in L1loc (R3 ; Mν (C)) and the existence of a constant C > 0 such that
(LGn)
∀k ∈ R3 , |k| ≤ 1,
∀k ∈ R3 , |k| ≥ 1,
and
|α| ≤ n :
and
|α| ≤ n :
k∂kα G(k)k ≤ C|k|n−1+µ−|α|
3
k∂kα G(k)k ≤ C|k|− 2 −δα,0 −µ .
We will make use of the condition (LGn) on G with n = 0, 1, 2. Note that (LGn)
implies (HGn). As for the Hamiltonian, there is nothing particular about dimension
3. The difference between (HGn) and (LGn) comes from having to absorb an
infrared singularity from the Boltzmann density
ρβ (k) =
1
eβ|k|
−1
,
which mix the left and right field components at positive temperature, β < ∞.
One could you use a different density, modifying (LGn) accordingly. See also
Remark 2.9.
We write
Gl (k) = G(k) ⊗ 1lK
and Gr (k) = 1lK ⊗ G(k).
7
(2.6)
At finite inverse temperature β we abbreviate
p
p
√
√
Gβl = 1 + ρβ Gl − ρβ G∗r and Gβr = 1 + ρβ Gr − ρβ G∗l .
(2.7)
#
#
#
It will also be convenient to write a#
l (k) = a (k) ⊗ 1lF and ar (k) = 1lF ⊗ a (k).
The interaction at positive temperature is Wβ (G) where
Wβ (G) := φl Gβl − φr Gβr .
Here the left and right fields are defined the obvious way. The zero and positive
temperature Liouvillean is thus densely defined, a priori on C L , as the operator sum
Lβ = L0 + Wβ (G) = L∞ + Iβ (G),
where
Iβ (G) = φl Gβl − Gl − φr Gβr − Gr .
(2.8)
That the Liouvilleans Lβ , 0 < β < ∞ are essentially self-adjoint on C L was proved
in [32, Lemma 3.2], cf. also [8, 13, 40], using Nelson’s commutator theorem [48,
Thm. X.37]. This requires that G can absorb a power of the dispersion |k|, which
is the source of the δα,0 term in the ultraviolet part of condition (LGn). We warn
the reader that the domain of Lβ may depend on both β and G, an issue that complicates the analysis of the operator. Proposition 2.4 2 below remedies this issue
somewhat.
We write N L = N ⊗ 1lF + 1lF ⊗ N for the number operator on F ⊗ F , and as
for N we use the same notation to denote 1lK⊗K ⊗ N L . Note that CN L C = N L .
We introduce the so called modular conjugation J, which is a conjugate linear
involution on HL given by
J := (C ⊗ C)E,
(2.9)
where E is the exchange operator defined on simple tensors by E(ψ ⊗ ϕ) = ϕ ⊗ ψ.
Here ψ, ϕ ∈ H. Clearly JL∞ J = −L∞ . Indeed, the identity holds on C L and
extends to D(L∞ ) since C L is an operator core for L∞ .
Computing as an identity first on C L we find
Jφl Gβl J = φr Gβr ,
and hence
JLβ J = −Lβ .
As above one should first verify the identity on C L and extend by continuity to
D(Lβ ). Consequently, we observe that the spectrum and pure point spectrum of
8
Lβ is reflection symmetric around 0. Furthermore the spectral resolution E β associated with Lβ satisfies E β (B) = JE β (−B)J and hence the absolutely and
singular continuous spectra of Lβ are also reflection symmetric.
We remark that there is a different way of representing the interaction Wβ (G)
which is more natural from the point of view of the underlying operator algebraic
framework. This representation makes use of the left and right Araki-Woods fields.
To construct these we introduce annihilation operators
q
q
AW
al (k) = 1 + ρβ (k) al (k) + ρβ (k) a∗r (k)
q
q
AW
ar (k) = 1 + ρβ (k) ar (k) + ρβ (k) a∗l (k)
and the Araki-Woods creation operators aAW∗
l/r (k) are now obtained by “taking
adjoints”. We can express Wβ (G) in terms of left and right Araki-Woods fields
φAW
l/r as follows
Wβ (G) = φAW
(Gl ) − φAW
r (Gr ),
l
(2.10)
where Gl/r , where introduced in (2.6).
We end the subsection with a proposition that permits us to work effectively
with standard Liouvilleans, despite domain problems. Its proof follows closely
arguments from [16], establishing similar statements for technically related operators.
Proposition 2.4. Suppose (LG0). The following holds
1 (L ) and the operator [N L , L ]◦ extends from D(N L ) by conti1. N L ∈ CMo
β
β
√
L
L
nuity to an element of B(D( N ); H ).
2. D(N L ) ∩ D(Lβ ) does not depend on β, nor on G.
3. C L is dense in D(N L ) ∩ D(Lβ ) with respect to the intersection topology.
Proof. To establish 1 we argue as in the verification of [16, Cond. 2.1 (2), cf. Sect. 5.5].
First observe that N L and L0 commute such that we can compute as a form on
the core C L
L
(N + 1)−1 , Lβ = (N L + 1)−1 Wβ (G) − Wβ (G)(N L + 1)−1 .
The right-hand side extends to a bounded operator, and since C L is dense in D(Lβ ),
the form [(N L + 1)−1 , Lβ ] defined on D(Lβ ) extends by continuity to a bounded
form on HL , coinciding with the closure of the right-hand side as a form on C L .
Hence N L ∈ C 1 (Lβ ).
Having established that N L is of class C 1 (Lβ ) we know that [N L , Lβ ] extends from the intersection domain D(N L ) ∩D(Lβ ) to a bounded form on D(N L ).
9
Hence, to compute this form it suffices to compute it on a core of N L . Compute as
a form on C L
i[N L , Lβ ] = Wβ (iG),
(2.11)
√
which extends from C L to B(D( N L ); HL ). This proves 1.
As for 2 we follow the proof of [16, Lemma 5.15]. Let T0 = L0 + i(N L + 1).
Since L0 and N L commute we clearly have D(T0 ) = D(L0 ) ∩ D(N L ) =: D0 .
b = L0 +
We now construct Lβ +i(N L +1) in two different ways. First define L
b + i(N L + 1) =: T1
Wβ (G) as a symmetric operator on D0 . Then T0 + Wβ (G) = L
is by [48, Corollary to Thm. X.48] a closed operator on D0 . Conversely we can
use [22, Thm. 2.25] to construct T2± = Lβ ± i(N L + 1) as closed operators on
b is
Dβ = D(Lβ ) ∩ D(N L ) with T2+∗ = T2− . Since C L ⊂ D0 we find that L
L
a symmetric extension of Lβ|C L . Hence, C being a core for Lβ , we find that
b ⊂ Lβ . This implies that T1 ⊂ T + =: T2 . Since T ± are both accretive we find
L
2
2
that T2 generates a contraction semigroup. To conclude the proof we only need to
show that the ρ(T1 ) ∩ ρ(T2 ) 6= ∅. But this follows from the Hille-Yosida theorem,
cf. [48, Thm. X.47a].
Finally we turn to 3. From what was just proved, together with the closed graph
theorem, we conclude that it suffices to prove that C L is dense in D0 with respect
to the norm
kψk0 = kN L ψkHL + kL0 ψkHL + kψkHL .
Since L0 and N L commute it suffices to show that one can approximate ψ ∈ D0
with ψ = 1l[N L ≤ n]ψ, for some n. Similarly, since L0 and N L commute with
ΓR := 1lK ⊗ Γ(1l[|k| ≤ R]) ⊗ 1lK ⊗ Γ(1l[|k| ≤ R]), it suffices to approximate states
ψ, non-zero in finitely many particle sectors, and satisfying ΓR ψ = ψ, for some
R > 0.
Let {ϕn }n∈N ⊂ C L be a sequence with kψ − ϕn kHL → 0 for n → ∞. Let χ ∈
∞
C0 (R3 ) satisfy 0 ≤ χ ≤ 1, χ(k) = 1 for |k| ≤ R, and χ(k) = 0 for |k| ≥ R + 1.
Then Γχ := 1lK ⊗ Γ(χ) ⊗ 1lK ⊗ Γ(χ) preserves C L and kψ − Γχ ϕn kHL → 0 for
n → ∞.
Now that both ψ and Γχ ϕn only have finitely many non-zero components all
supported inside a box of side length R + 1, one can easily verify that
kψ − Γχ ϕn k0 → 0,
for n → ∞.
This completes the proof.
We remark that Lβ is presumably not of class C 1 (N L ), cf. Remark 2.2.
10
2.3 Jakšić-Pillet Gluing
We proceed to discuss a unitarily equivalent form of the Liouvillean obtained by
so called the Jakšić-Pillet gluing procedure, cf. [13, 32]
But first we pass to polar coordinates on the Hamiltonian level. Define a unitary
transform Tl : h → h̃l := L2 ([0, ∞)) ⊗ L2 (S 2 ) by the prescription
(Tl f )(ω, Θ) = ωf (ωΘ).
Denote by Fel = Γ(h̃l ) the Fock space in polar coordinates. The subscripts l is for
later use and refers to the left component in the tensor product HL = H ⊗ H. The
twiddle indicates an object represented in polar coordinates for the Hamiltonian,
and after gluing for the Liouvillean.
The coupling in polar coordinates becomes
e
G(ω,
Θ) := ωG(ωΘ)
and the Hamiltonian takes the form
e = 1lK ⊗ Γ(Tl ) H 1lK ⊗ Γ(Tl ) = K ⊗ 1l e + 1lK ⊗ dΓ(ω) + φ(G),
e
H
Fl
e = D(1lK ⊗
a priori as an identity on C0∞ ([0, ∞) ⊗ C ∞ (S 2 ) and extended to D(H)
dΓ(ω)) by continuity.
To deal with the Liouvillean we similarly need a map Tr : h → h̃r := L2 ((−∞, 0])⊗
L2 (S 2 ), defined by (Tr f )(ω, Θ) = (Tl f )(−ω, Θ). Put Fer = Γ(h̃r ). This sets up a
unitary transformation
T : h ⊕ h → h̃ := L2 (R) ⊗ L2 (S 2 ),
by the construction
(T (f, g))(ω, Θ) = 1l[ω ≥ 0](Tl f )(ω, Θ) + 1l[ω ≤ 0](Tr g)(ω, Θ).
Using the canonical identification I : Γ(h⊕h) → F ⊗F we construct a unitary
map
e L := K ⊗ K ⊗ Fe,
U : HL → H
where Fe = Γ(L2 (R) ⊗ L2 (S 2 )). The map U is defined on simple tensors by
U (v ⊗ η ⊗ w ⊗ ξ) = v ⊗ w ⊗ Γ(T )I ∗ (η ⊗ ξ)
and extended to HL by linearity. Here v, w ∈ K and ξ, η ∈ F . As an alternative
core we take
CeL = K ⊗ K ⊗ Γfin C0∞ (R) ⊗ C ∞ (S 2 ) .
11
Let LK = K ⊗ 1lK − 1lK ⊗ K as an operator on K ⊗ K.
In the new coordinate system, we can write the interaction Wβ (G) as a field
operator as follows. First, the zero temperature interaction is
e ∞ := 1l[|ω ≥ 0]G
e l (ω, Θ) − 1l[ω ≤ 0]G
e r (−ω, Θ),
G
(2.12)
e l/r (ω, Θ) = ωGl/r (ωΘ), cf. (2.6). With this construction we have U (φl (Gl )−
where G
e ∞ ). The computation is easily done on CeL and extended by conφr (Gr ))U ∗ = φ(G
e
tinuity to D(φ(G)). At finite temperature, the interaction reads
q
q
e β (ω, Θ) := 1 + ρeβ G
e ∞ + ρeβ G
e∗ ,
G
(2.13)
∞,R
e ∞,R (ω, Θ) = G
e ∞ (−ω, Θ) is the reflected glued coupling, and
where G
ρeβ (ω, Θ) = ρβ (ωΘ) =
1
eβ|ω|
−1
.
Recalling (2.7), we observe that we similarly have U (φl (Gβl ) − φr (Gβr ))U ∗ =
e β ).
φ(G
eβ
As observed in [32] we have the following alternative representation of G
where
eβ =
G
ω
1 − e−βω
1
2
bl −
G
ω
βω
e −1
1
2
br ,
G
√
√
b l (ω, Θ) = 1l[ω ≥ 0] ωG(ωΘ) + 1l[ω ≤ 0] −ωG(−ωΘ)∗ ⊗ 1lK
G
√
√
b r (ω, Θ) = 1lK ⊗ 1l[ω ≥ 0] ωG(ωΘ) + 1l[ω ≤ 0] −ωG(−ωΘ)∗ .
G
(2.14)
(2.15)
This form of the interaction mirrors the Araki-Woods representation (2.10).
Remark 2.5. The representation (2.13) allows us to easily observe that under the
e β and its first n derivatives are square
assumption (LGn), the ultraviolet part of G
integrable, whereas (2.14) allows us to conclude the same for the infrared region.
♦
We can now write down the standard Liouvillean in the new coordinate system
as
e β = LK ⊗ 1l e + 1lK⊗K ⊗ H
e ph + φ(G
e β ),
L
F
e ph = dΓ(ω). Note that ω denotes both a real number and multiplication by
with H
the identity function in L2 (R).
12
e β is essentially self-adjoint on CeL .
Again, by Nelson’s commutator theorem, L
e β are unitarily equivalent through U . As an identity on
We observe that Lβ and L
L
e
eβ − G
e ∞ ), with L
e∞ = L
e 0 + φ(G
e ∞ ) and L
e0 =
C we have Lβ = L∞ + φ(G
e ph . These operators are also essentially self-adjoint on CeL
LK ⊗ 1lFe + 1lK⊗K ⊗ H
and their closures are unitarily equivalent with the appropriate untwitled objects.
e = U N L U ∗ = dΓ(1l ), where the
In the glued coordinate system we write N
h̃
second quantization is here performed in Fe.
The statements 1 and 2 in the following corollary to Proposition 2.4 are an
immediate consequence of Proposition 2.4 1 and 2. The item 3 however is not, but
it can be proved by an argument identical to the one employed at the end of the
proof above.
Corollary 2.6. Suppose (LG0). The following holds
e ∈ C 1 (L
e β ) and the operator [N
e, L
e β ]◦ extends from D(N
e ) by continuity
1. N
Mo
p
e ); H
e L ).
to an element of B(D( N
e ) ∩ D(L
e β ) does not depend on β, nor on G.
2. D(N
e ) ∩ D(L
e β ) with respect to the intersection topology.
3. CeL is dense in D(N
We remark that it is a consequence of Proposition 2.4 and the above corollary
e β are strongly continuous in
that, supposing (LG0), the resolvents of Lβ and L
β ∈ (0, ∞] and G, using the normp
kGk′0 = k(1 + |k|−1/2 )Gk. Indeed, it suffices
e ) where we compute
to prove strong convergence on D( N
eβ − G
e′ ′ )(Lβ (G)−z)−1 .
(Lβ (G)−z)−1 −(Lβ ′ (G′ )−z)−1 = (Lβ ′ (G′ )−z)−1 φ(G
β
p
−1 : D( N
e) →
Here
we
used
[16,
Lemma
3.3]
to
observe
that
(L
(G)
−
z)
β
p
e ). The result now follows by observing that
D( N
G
eβ − G
e ′ ′ = 0.
lim
β
(β ′ ,G′ )→(β,G)
Norm continuity, even in β, of the resolvent is probably false but an argument is
lacking. This means that while σ(L∞ ) = R, cf. [47, Thm. VIII.33], we cannot
a priori exclude that the spectrum of Lβ could collapse for β < ∞ to become a
proper subset of R, cf. the discussion around [47, Thm. VIII.24].
Instead we proceed as for the Hamiltonian, via a pull through formula:
p
e ) we
Proposition 2.7. Suppose (LG0). For any β > 0, z ∈ C\R and ψ ∈ D( N
2
2
L
e )-identity
have as an L (R × S ; H
e β −z)−1 ψ = (L
e β +ω−z)−1 a(ω, Θ)ψ−(L
e β +ω−z)−1 (G
e β (ω, Θ)⊗1l e )ψ.
a(ω, Θ)(L
F
13
Using this pull through formula we obtain
e β ) = R.
Theorem 2.8. Suppose (LG0). For any β > 0 we have σ(Lβ ) = σ(L
We omit proofs of Proposition 2.7 and Theorem 2.8 since they are verbatim the
same as for Proposition 2.1 and Theorem 2.3, keeping in mind
p Corollary 2.6, in
e
e ), and that CeL is
particular the consequence that resolvents of Lβ preserves D( N
eβ .
a an operator core for L
The two HVZ-type theorems, Theorems 2.3 and 2.8, will play no role in the
notes apart from clarifying the general spectral picture.
Remark 2.9. Our results on the standard Liouvillean will mostly be proved in the
Jakšić-Pillet glued coordinates. Since only ω-derivatives will play a role, this ale β instead of G. This
lows us to formulate slightly weaker assumptions using G
improvement is in general largely irrelevant, hence the present formulation with
(LGn).
More importantly, for couplings G on a special form, one can due to the representation (2.14) allow for interactions at positive temperature far more singular
than what is permitted by (LGn). To make this precise assume G takes the form
G(k) = |k|−1/2 g(k)G0 ,
where G0 ∈ Mν (C) is self-adjoint G∗0 = G0 , and g : R3 → R. Define
ĝ(ω, Θ) = 1l[ω ≥ 0]g(ωΘ) + 1l[ω ≤ 0]g(−ωΘ).
b l = ĝG0 and G
b r = ĝG0 , cf. (2.15). Hence we see
Then we can represent G
e
that differentiability of Gβ is governed by that of ĝ. For the spin-boson model
g is a form factor (or ultraviolet cutoff), e.g. constant near 0 or perhaps of the
2
2
form e−k /Λ to take some popular choices. Here ĝ will be constant across the
2
2
singularity at ω = 0, or equal to e−ω /Λ for the other choice.
For this class of models we can reformulate replacements for (LGn). Let n ∈
N0 . There exists ĝ : R × S 2 → C and G0 ∈ Mν (C) self-adjoint such that ĝ admits
n distributional ω derivatives in L1loc (R × S 2 ) and such that
1
(LGn′ )
G(ω, Θ) = |ω|− 2 ĝ(ω, Θ) G0
∀ωΘ ∈ R3 , |ω| ≤ 1,
∀ωΘ ∈ R3 , |ω| ≥ 1,
and
and
j≤n:
j≤n:
|∂ωj ĝ(ω, Θ)| ≤ C|ω|n−1+µ−j
|∂ωj ĝ(ω, Θ)| ≤ C|ω|−1−δj,0 −µ .
This type of condition was used in [14, 18].
Finally we remark that it was observed and utilized in [18], that the Jakšić-Pillet
gluing is not canonical in that one can glue the two reservoirs together at ω = 0,
14
twisting one of them with a phase. This allows one to consider ĝ as complex valued
and then pick the gluing phase such that ĝ(ωΘ) and ĝ(−ωΘ) fit together seamlessly
across ω = 0. In fact, one can in this way also allow for singular behavior of the
form |k|1/2 at zero, and not just |k|−1/2 . This would require an extra twist by the
angle π corresponding to a sign change across zero.
♦
2.4 Multiple Reservoirs
We have made a choice, in the name of concreteness to focus on finite dimensional
quantum systems coupled to a massless scalar field (in three dimensions) and their
thermal Liouvilleans.
Our methods and indeed theorems however have validity beyond this particular
choice. We single out here the case of multiple reservoirs at possibly different
inverse temperatures β~ = (β1 , . . . , βq ).
The easiest way to observe that the results of these notes carry over to the case
of multiple reservoirs is to replace h = L2 (R3 ) by L2 (R3 × {1, . . . , q}) ∼ hq , q
being the number of reservoirs. The dispersion becomes ω(k, j) = |k| (or |k|1lCq ).
Given q couplings G1 , . . . Gq all satisfying the same sets of conditions, one can
construct a coupling for the multi-reservoir system by setting G(k, j) = Gj (k).
~
As for the standard Liouvillean one should replace Gβl/r by functions Gβl/r (·, j) =
β
e~.
Gl/rj (·, j). Similarly for G
β
Weak-coupling as well as high and low-temperature results remain valid if all
coupling, respectively temperatures, are taken into the same regime.
Only one type of result here does not extend to the case of multiple reservoirs,
and that is the existence/non-existence results for eigenvalues of Lβ discussed in
Subsect. 3.4, which make critical use of the modular structure of the thermal Liouvilleans. In fact, if two inverse temperatures are distinct, at weak coupling and
under a suitable non-triviality condition on G one has σpp (Lβ~ ) = ∅, cf. [14,
Thm. 7.17].
One could also replace the thermal density ρβ by other densities and a number
of our results (except for low temperature statements) remain valid. However, the
reader doing that would have to reformulate the condition (LGn) where the 1/|k|
singularity of ρβ is built in. The reader can consult [14, 15] for discussions of other
models.
Finally we remark that essentially what we exploit is the Jakšić-Pillet glued
representation of the standard Liouvillean and presumably one could rephrase everything in this abstract setup. See also [14, Sect. 8].
The reader can consult [34, 35, 41, 42] for papers devoted to multiple reservoirs. Three of these papers consider also the non-selfadjoint C-Liovillean, which
15
seems to be a more natural object when considering non-stationary steady states.
2.5 Open Problems I
There are not that many serious problems pertaining to the material from this section. We did mention two related conjectures regarding the standard Liouvillean,
while not in itself of great interest, resolving them would serve to clarify the picture:
Problem 2.1. Clarify to what extend the domain of the standard Liouvillean Lβ is
β and G dependent.
Problem 2.2. Verify that, as conjectured, the resolvents of the Liouvillean are not
norm continuous in β and G.
As a final topic, we discuss the ultraviolet singularity
of the models. For e.g.
p
the spin-boson model, the coupling G goes as 1/ |k| for large momenta, which
is more singular than what we can deal with. It is well-known that the Nelson
(and the polaron) model is renormalizable, but this is due to a regularizing effect
stemming from the small system, in that the Laplacian allows for control of the
ultraviolet contributions [3, 38, 44]. Indeed, we do not expect that the model has
a meaningful ultraviolet limit and it should not be a relevant question since it is
a model describing low energy/momentum phenomena only. Having said that, it
would still be undesirable if the choice of (a reasonable) cutoff would influence
whether or not the Liouvillean admits non-zero eigenstates, and if it does, will the
point spectrum, being related to energy differences, have an ultraviolet limit. This
is an underlying issue that will not play a role in these notes since our focus will
be on the more physically relevant infrared regime.
3 Bound States
In this section we study the basic properties of bound states. The key is the following formal computation
hψ, i[H, A]ψi = 0,
whenever ψ is a bound state for H and A is some auxiliary operator. Choosing A
such that the commutator i[H, A] contains a positive operator N and a remainder
controllable either by H or some fractional power of N , will imply - at least formally - that ψ is in the form domain of N . In our case, the operator N will be the
number operator N (or N L for the Liouvillean).
It turns out to be a surprisingly delicate question to establish such a bound rigorously for the standard Liouvillean, but for the Hamiltonian it is fairly straightforward. The first argument of this type is for the Hamiltonian and is due to Skibsted
16
[49], and for the Liouvillean it goes back to Fröhlich and Merkli [19], cf. also [20].
The result we present here for the Liouvillean improves on the theorem of Fröhlich
and Merkli.
As a consequence of such number bounds we will be able to establish virial
theorems for the Hamiltonian and the Liouvillean.
3.1 Number Bounds at Zero Temperature
As for A we make the choice
A = dΓ(a),
with
k
k
i
a=
· ∇k + ∇k ·
,
(3.16)
2 |k|
|k|
the generator of radial translations. Note that a should be viewed as the closure of
a restricted to C0∞ (R3 \{0}) and that a is a maximally symmetric operator, but not
self-adjoint. Since H is not of class C 1 (A), we cannot directly make sense out of
the formal computation above.
Instead we introduce a family of regularized conjugate operators An = dΓ(an )
with
(
)
k
k
i
p
· ∇k + ∇k · p
an =
.
2
|k|2 + n−1
|k|2 + n−1
The an ’s, constructed as closures from C0∞ (R3 ), are self-adjoint and H ∈ C 1 (An )
for all n, provide (HG1) is assumed. This construction goes back to [49] and was
used also in [23].
Let ψ be a bound state for H, i.e. Hψ = Eψ for some E ∈ R. It is now a
consequence of the standard Virial Theorem, cf. [12, 21], that hψ, i[H, An ]◦ ψi =
0. Computing the commutator we find
|k|
− φ(ian G).
i[H, An ]◦ = dΓ p
|k|2 + n−1
2
3
Note that assuming (HG1)
p we have an G ∈ L (R ; Mν (C))). From the estimate
2
−1
(2.3), applied with |k|/ |k| + n in place of |k|, we get
2
1
1
|k|
1
i[H, An ]◦ ≥ dΓ p
− C (|k|2 + n−1 ) 4 |k|− 2 an G .
2
|k|2 + n−1
To check for the finiteness and uniform boundedness
√ of the norm on the right-hand
side we write an = √ 2k −1 · i∇k + 2i div(k/ k 2 + n−1 ) and estimate
|k| +n
(|k|2 + n−1 ) 41 |k|− 21 an G ≤ ∇G + C G/|k|,
17
(3.17)
p
for some n-independent constant C. Since n → |k|/ |k|2 + n−1 is monotonously
increasing towards 1 we conclude from Lebesgue’s theorem on monotone convergence the following:
Theorem 3.1. Suppose (HG1). There exists
√ a C > 0 such that for any normalized
bound state ψ ∈ H of H we have ψ ∈ D( N ) and
√
N ψ ≤ C k∇Gk + kG/|k|k .
Since N commutes with the conjugation C, we observe that the same theorem
holds for bound states of H c .
That the constant C in the theorem above can be chosen uniformly in E, is a
consequence of K being finite dimensional. For e.g. the confined Nelson model,
this would be false since one will need a resolvent of K to bound the relevant aG.
This is however a mute point, since we in Subsect. 4.3 will prove that H does not
have high energy bound states!
In fact if one assumes in addition (HG2) one can do better and get ψ ∈ D(N )
using [16]. This is however a much deeper result and will not play a role in these
notes.
3.2 Number Bounds at Positive Temperature
For the standard Liouvillean L∞ at zero temperature we observe that since eigenstates are on the form ψ ⊗ ϕ, with ψ, ϕ eigenstates of H and√H c respectively, they
L
are due to Theorem 3.1 automatically
p in the domain of D( N ). Hence, eigene ∞ are in the domain of N
e . The situation at positive temperature is a
states of L
good deal more subtle.
We begin with a key technical lemma, which enables us to compute commutators. The proof follows closely a similar argument from [16, Proof of Cond. 2.1 (3)].
Before stating the lemma we need some notation. Let m ∈ C 1 (R) be realvalued and bounded with bounded derivative. Put
d
d
i
m
+
m ⊗ 1lL2 (S 2 )
(3.18)
ãm =
2
dω dω
and
em = 1lK⊗K ⊗ dΓ(ãm ).
A
(3.19)
em are essentially self-adjoint on
We leave it to the reader to argue that ãm and A
∞
∞
2
2
L
e
C0 (R) ⊗ C (L (S )) and C respectively.
18
Lemma 3.2. Suppose (LG1). Then
e β , (A
em − z)−1 ]ϕ = − ψ, (A
em − z)−1 L
e ′ (A
em − z)−1 ,
ψ, i[L
β
p
e ) ∩ D(L
e β ) and z ∈ C with Imz 6= 0. Here
for all ψ, ϕ ∈ D( N
defined as a form on D(
e ′ = 1lK ⊗ dΓ(m) − φ(iãm G
eβ )
L
β
p
e ).
N
Remark 3.3. We first observe that the expression in the lemma
p makes sense. Since
−1
e . By boundedness
e
e
e
Am and N commute, (Am − z) preserves the domain of N
′
′
e is well-defined as a form on
of m and m together with Remark 2.5 we see that L
β
p
e
D( N ).
eβ ) ∩
Proof. By [16, Remark 3.5], it suffices to check the identity for ψ, ϕ ∈ D(L
e ), which by Corollary 2.6 equals D(L
e 0 ) ∩ D(N
e ). On this domain L
e β can be
D(N
e
e
written as the operator sum L0 + φ(Gβ ). Hence it suffices to prove that
e 0 , (A
em − z)−1 ]ϕ = − ψ, (A
em − z)−1 1lK ⊗ dΓ(m)(A
em − z)−1 ϕ ,
ψ, i[L
e β ), (A
em − z)−1 ]ϕ = ψ, (A
em − z)−1 φ(iãm G
eβ )(A
em − z)−1 ϕ ,
ψ, i[φ(G
(3.20)
e
e
for all ψ, ϕ ∈ D(L0 ) ∩ D(N ) and z ∈ C with Imz 6= 0. The second identity in
eL
(3.20) can easily be verified
pfor ψ, ϕ ∈ C from which it extends by density since
e β ) and φ(iãm G
e β ) are N
e -bounded, cf. Remark 2.5.
φ(G
As for the first identity in (3.20) one should first observe that all objects pree = n for some n. Hence it suffices
serve particle sectors, i.e. sectors with N
to establish the identity for ψ, ϕ being n-particle states. Observe that dΓ(n) (ω)
1 (dΓ(n) (ã )), indeed; i[dΓ(n) (ω), dΓ(n) (ã )]◦ = dΓ(n) (m) is a
is of class CMo
m
m
bounded operator on F (n) , the n-particle sector. Hence the identity
ψ, 1lK⊗K ⊗ i[dΓ(n) (ω), (dΓ(n) (ãm ) − z)−1 ]ϕ
= − ψ, 1lK⊗K ⊗ (dΓ(n) (ãm ) − z)−1 dΓ(n) (m)(dΓ(n) (ãm ) − z)−1 ϕ
holds for z with |Imz| ≥ σn for some σn chosen such that (dΓ(n) (ãm ) − z)−1
preserves D(dΓ(n) (ω)) inside the n-particle sector, cf. [43, Prop. II.3]. By the
unique continuation theorem, the identity then holds for all z with Imz 6= 0.
We are now ready to state and prove our improvement of the Fr öhlich-Merkli
number bound. For comparison, we require one less commutator reflected in an
19
improvement by one power of |k| in the infrared behavior of G. It is however still
one commutator more than what was needed for the Hamiltonian. It is unclear if
this is just a technical issue.
√
Theorem 3.4. Suppose (LG2). Let ψ be an eigenstate of Lβ . Then ψ ∈ D( N L ).
e β ) be an eigenstate for L
e β . It suffices to prove that ψ ∈
Proof. Let ψ ∈ D(L
1/2
e ). We can assume without loss of generality that the eigenvalue is zero, i.e.
D(N
e
Lβ ψ = 0.
Denote by ã = ãm≡1 the generator of translations, cf. (3.18). Similarly we
e=A
em≡1 = 1lK⊗K ⊗ dΓ(ã). Note that A
e commutes with N
e.
abbreviate A
1
e
e
e
eβ ) ∩
Since N by Corollary 2.6 is of class C (Lβ ) we have In (N )ψ ∈ D(L
e ), cf. [43, Prop. II.2]. Here
D(N
e ) = n(N
e + n)−1
In ( N
e ) = 1l e L .
satisfies s − lim In (N
H
n→∞
(3.21)
e )ψ. With the choice m = 1 we get
Put ψn = In (N
e′ = N
e − φ(iãG
e β ),
L
β
e ). We can thus compute using Lemma 3.2
as a self-adjoint operator with domain D(N
for m ∈ N and z ∈ C, with Imz 6= 0,
e β , (A/m
e
ψn , i[L
− z)−1 ]ψn
1
e ′ (A/m
e
=−
ψn , (A/m − z)−1 L
− z)−1 ψn
β
m
1
e ′ (A/m
e
ψn , L
− z)−2 ψn
=
β
m
i e
e β )(A/m
e
− 2 ψn , (A/m
− z)−1 φ(ã2 G
− z)−2 ψn .
m
e ′ is of class C 1 (A)
e with [L
e ′ , A]
e ◦ = iφ(ã2 G
e β ).
In the last equality we used that L
Mo
β
β
On the other hand we can undo the commutator on the left-hand side and commute
e β through In (N
e ) to get
L
e β , (A/m
e
ψn , i[L
− z)−1 ]ψn
e β )(N
e + n)−1 (A/m
e
e
e + n)−1 φ(iG
e β ) ψn .
= − ψn , φ(iG
− z)−1 + (A/m
− z)−1 (N
Here we used Corollary 2.6 and a twidled version of (2.11).
Let g ∈ C ∞ (R) be identical to t for |t| ≤ 1, monotonously
√ increasing and
constant outside a ball of radius 2. Suppose in addition that g ′ is smooth. We
will furthermore require that
∀t ∈ R :
tg ′′ (t) ≤ 0.
20
(3.22)
Let g̃ denote an almost analytic extension of g. Abbreviating gm (t) = mg(t/m)
we get
e β )(N
e + n)−1 gm (A)
e + gm (A)(
e N
e + n)−1 φ(iG
e β ) ψn
− ψn , φ(iG
e ′ g ′ (A)ψ
e n
= ψn , L
β m
Z
i
¯
e
eβ )(A/m
e
∂g̃(z)
ψn , (A/m
− z)−1 φ(ã2 G
− z)−2 ψn dz.
−
mπ C
√
We estimate the left-hand side to be O(m/ n) and the second term on the right√
hand side is O( n/m), cf. Remark 2.5. Hence we get
√ m
n
e ′ g ′ (A)ψ
e n ≤C √
ψn , L
+
(3.23)
β m
m
n
p
for some C > 0. Put h(t) = g ′ (t). Then h ∈ C0∞ (R) and defining hm (t) =
e 2 = g ′ (A).
e Let h̃ be an almost analytic extension of h.
h(t/m) we find hm (A)
m
Then
e ′ g ′ (A)ψ
e n = ψn , hm (A)
eL
e ′ hm (A)ψ
e n + ψn , [L
e ′ , hm (A)]h
e m (A)ψ
e n .
ψn , L
β m
β
β
e L
e ′ ](N
e + 1)−1/2 = [hm (A),
e φ(iãG
eβ )](N
e + 1)−1/2 is of the
Observe that [hm (A),
β
order 1/m and hence
√ n
′ ′
′
e
e
e
e
e
ψn , Lβ gm (A)ψn = ψn , hm (A)Lβ hm (A)ψn + O
.
(3.24)
m
e′ ≥ N
e /2 − C ′ , for some C ′ > 0, we get from (3.23) and (3.24)
Finally using that L
β
that
√ m
n
eN
e hm (A)ψ
e n ≤C √
+
ψn , hm (A)
n
m
for some C > 0 and all n, m ≥ 1.
We now pick n = m2 such that we obtain the bound
e )2 N
e hm (A)ψ
e
e m2 ( N
≤ 2C
ψ, hm (A)I
uniformly in m.
e e
Let E (N ,A) be the joint spectral resolution on N0 × R induced by the two
e and A.
e Then
commuting operators N
Z
e ,A)
e
nm4
(N
e )2 N
e hm (A)ψ
e
e m2 ( N
(n, t).
dEψ
=
hm (t)2
ψ, hm (A)I
2
2
(n + m )
N0 ×R
21
4
nm
Since hm (t)2 (n+m
2 )2 → n monotonously as m → ∞ we conclude using the
R
e ,A)
e
(N
monotone convergence theorem that N0 ×R n dEψ
(n, t) < ∞. Here we used
(3.22) to ensure that m → hm (t) is monotonously increasing towards 1. Being a
joint spectral resolution we have
Z
Z
e ,A)
e
e
(N
n dEψN (n),
n dEψ
(n, t) =
N0 ×R
N0
e
e . Hence hψ, N
e ψi < ∞ and we are
where E N is the spectral resolution for N
done.
3.3 Virial Theorem’s
Having established the number bounds we can now formulate and prove two virial
theorems
Let m ∈ C 1 (R) be real-valued and bounded with bounded derivative as in
the previous subsection. Given such a function m we can construct a maximally
symmetric operator by the prescription
m(|k|)k
i m(|k|)k
· ∇k + ∇k ·
am =
2
|k|
|k|
on C0∞ (R3 \{0}). This gives rise to a maximally symmetric operator Am = dΓ(am )
on H. We write, supposing (HG1),
′
Hm
= dΓ(m(|k|)) − φ(iam G),
√
as a form on D( N ), cf. (2.2). If inf m(ω) > 0, then H ′ is self-adjoint on D(N ).
We have
Theorem 3.5. Suppose (HG1). Let ψ ∈ H be a bound state for the Hamiltonian
′ ψi = 0.
H. Then hψ, Hm
Proof. Note that the expectation value is meaningful due to the number bound in
Theorem 3.1.
√
Replace m by a regularizing function mn (r) = m(r)r/ r2 + n−1 as in the
proof of the number bound in Subsect. 3.1. Then the associated amn is self-adjoint
and so is Amn = dΓ(amn ). Furthermore, H is of class C 1 (Amn ) for all n. We can
compute the commutator i[H, Amn ]◦ = dΓ(mn (|k|)) − φ(iamn G). By the usual
virial theorem, cf. [12, 21], together with (3.17), Theorem 3.1 and Lebesgue’s
dominated convergence theorem we conclude the proof.
22
em from
To deal with the standard Liouvillean we use the observables ãm and A
(3.18) and (3.19). Write, supposing now (LG1),
e ′ = 1lK⊗K ⊗ dΓ(m(ω)) − φ(iãm G
e β ),
L
β
p
e ). Again, if inf m(ω) > 0,
which by Remark 2.5 is a well-defined form on D( N
′
e ). Define
e is self-adjoint on D(N
then L
β
e′ U ,
L′β = U ∗ L
β
√
which is a well-defined form on D( N L ). Under the (LG1) assumption we can
compute
L′β = 1lK⊗K ⊗ dΓ(m(|k|) ⊗ 1lF + 1lF ⊗ dΓ(m(|k|) − φl (iam Gβl ) + φr (iam Gβr ).
We warn the reader that if one follows Remark 2.9 and imposes an (LG1’) assumption instead of (LG1), then am Gβl/r may not be well-defined.
Theorem 3.6. Suppose (LG2). Let ψ ∈ HL be a bound state for the standard
Liouvillean Lβ , at inverse temperature 0 < β ≤ ∞. Then hψ, L′β ψi = 0.
Proof. First of all we note that the expectation value is meaningful due to Theorem 3.4. Secondly, it suffices to prove the theorem in the glued coordinates where
eβ .
e ′ and ψ is an eigenstate for L
L′β is replaced by L
β
e β . Using the notation In (A
em ) = in(A
em + in)−1 ,
Let ψ be a bound state for L
cf. also (3.21), we write
em In (A
em ) = in1l eL + n2 (A
em + in)−1 .
Bn = A
H
em ) = 1l eL . We compute using
Then Bn is bounded for all n and limn→∞ In (A
H
p
e)
Lemma 3.2 as a form on D( N
e β , Bn ]ψ = ψ, In (A
em )L
e ′ In ( A
em )ψ .
0 = ψ, i[L
β
e commutes with In (A
em ), we can - keeping Theorem 3.4 in mind - take the
Since N
limit n → ∞ and conclude the theorem.
The theorem of course remains true if we pass to the Jakšić-Pillet glued opere β . While the proof given above is at least formally identical to a standard
ator L
proof of the usual virial theorem, the reader should keep in mind that it relies on
the non-trivial Lemma 3.2 and Theorem 3.4.
The virial theorem’s are the tools that will allow us to deduce statements about
non-existence, local finiteness, and finite multiplicity, for eigenvalues given a so
called positive commutator estimate. This is the subject of Sect. 4
23
3.4 A Review of Existence and Non-existence Results
The first theorem we highlight is due to Gérard [24, Thm. 1] and establishes existence of a ground state for the Hamiltonian H under an (HG1) condition. Subsequently some improvements appeared in [10, 45].
Theorem 3.7. Assume (HG1). Then the bottom of the spectrum Σ of H is an
eigenvalue.
In a somewhat surprising recent development Hasler and Herbst proved that the
Spin-Boson model, cf. Remark 2.9, admits a ground state if the coupling is sufficiently weak [28]. They used the renormalization group method of Bach, Fröhlich
and Sigal [7]. See also Problems 3.3 and 3.4 in the following subsection.
The following beautiful theorem, due to Derezinski, Jaksic and Pillet establishes the existence of a β-KMS vector, which is in particular an eigenvector of
Lβ with eigenvalue zero. See [14, Thm. 7.3] and [15, Appendix B]. This improves
on an earlier result of Bach, Fr öhlich and Sigal [8, Thm. IV.3], who required more
infrared regularity.
Theorem 3.8. Suppose (LG0). Then for any inverse temperature 0 < β < ∞ the
standard Liouvillean Lβ has a β-KMS vector sitting in the kernel.
It is worth noting that although the above theorem mirrors G érard’s result for
the Hamiltonian, it holds true for more singular interactions. In particular, one can
not rule out a situation where H has no ground state, but Lβ has a β-KMS vector
in its kernel. Indeed, this situation actually occurs in the ν = 1 case. Here the
Pauli-Fierz Hamiltonian is of the type considered by Dereziński in [11], where it is
referred to as a van Hove Hamiltonian. If we consider
1
G(k) = |k|− 2 ĝ(k),
with ĝ ∈ C0∞ (R3 ) real-valued playing the role of an ultraviolet cutoff. We put
ĝ(0) = 1 such that the infrared behavior is captured by |k|−1/2 . It satisfies (LG0)
needed for Theorem 3.8, but not (HG1) needed for Theorem 3.7.
With this coupling the Hamiltonian becomes of infrared type II, again referring
to the terminology of [11], and does not admit a ground state. The ground state
−3/2 ĝ)
should be the coherent state eiφ(i|k|
|0i, but this is not in the Fock-space since
−3/2
2
3
|k|
ĝ 6∈ L (R ). To see what happens with the standard Liouvillean we observe
that for ν = 1 (and real ĝ) we have
Gβl = Gβr =
p
1 + ρβ −
24
√ − 21
ρβ |k| ĝ.
Expanding ρβ around k = 0 we see that
Gβl/r
p
1 + ρβ −
√
β
∼
2
p
√
ρβ ∼ β|k|/2. Hence
(3.25)
at k = 0. Hence we can diagonalize the Liouvillean with a tensor product of Weyl
operators as follows. Put
−1 Gβ )
l
V = eiφ(i|k|
−1 Gβ )
r
⊗ eiφ(i|k|
,
which due to (3.25) is a well-defined unitary operator. Then V ∗ Lβ V = L0 and
V (|0i ⊗ |0i) is the only eigenstate and in particular the β-KMS state. Note that the
energy shift one gets for the Hamiltonian does not occur here, since the shift from
the left and right components cancel each other out.
The final result we discuss in this subsection is a consequence of Theorem 3.8
and a theorem of Jadczyk [31], which has as a consequence that existence and
simplicity of the 0 eigenvalue for the standard Liouvillean implies non-existence
of non-zero eigenvalues! We refer the reader to the short and very elegant paper
[33] for details, which are entirely operator algebraic in nature.
Theorem 3.9. Suppose (LG0). Let 0 < β < ∞ and suppose that 0 is a simple
eigenvalue for Lβ . Then σpp (Lβ ) = {0}.
3.5 Open Problems II
As the reader may have observed, the bottleneck for applying the virial theorem to
the standard Liouvillean is the number bound Theorem 3.4, where we compared
with the Hamiltonian case Theorem 3.1 need much stronger assumptions. This is
in particular unfortunate since the positive commutator estimates we establish in
the following section hold under an (LG1) assumption, not the (LG2) assumption
needed for the number bound.
Problem 3.1. Can the number bound in Theorem 3.4 be established under an (LG1)
condition, or some other condition truly weaker than (LG2).
The author does not know one way or the other what the answer may be this
problem. We remark that although, the number bound is a bottleneck viz a viz
the structure of the point spectrum, the (LG2) condition is what one would expect
for a limiting absorption principle to hold, given a positive commutator estimate.
Hence, from a broader picture the (LG2) condition will appear anyway.
The proof of the number bound Theorem 3.4 did not make essential use of the
small system being finite dimensional. Hence we expect the theorem to remain
true also for confined small systems, like the standard Liouvillean for the confined
Nelson model.
25
Problem 3.2. Extend Theorem 3.4 to the case where the small system K is not
necessarily finite dimensional.
As mentioned in Subsect. 3.4, Hasler and Herbst established in [28] the existence of an interacting ground state for the spin-boson model with physical infrared
singularity |k|−1/2 , provided the coupling is sufficiently weak. This result came as
a complete surprise to the author, since it is contrary to the solvable model with
K = C and the confined Nelson model [10, 11, 30, 39, 46]. Furthermore it goes
beyond what was considered the natural borderline established in [24], cf. also
[6, 10, 45]. In fact there has been speculation that gauge invariance of the minimally coupled model was responsible for the existence result of Griesemer-LiebLoss [26, 37], something that was however debunked by Hasler-Herbst [27, 29]
who proved that existence of a ground state, at weak coupling, remains true even
after dropping the quadratic term in the minimally coupled model, thus breaking
gauge invariance.
The |k|−1/2 infrared behavior of G is sometimes called the “ohmic case”, a
terminology we use below.
Problem 3.3. Does there exists a critical coupling at which the ground state seize to
exist for the spin-boson model considered by Hasler and Herbst? Or does a ground
state exist for all couplings?
Problem 3.4. Characterize the properties of ohmic G that ensures existence of a
ground state for H in the weak coupling regime. As a simpler problem, consider
G’s on the form G(k) = |k|−1/2 ĝ(k)G0 as discussed in Remark 2.9.
For the thermal standard Liouvillean, one has existence of a β-KMS vector in
the kernel of Lβ at all values of β, cf. Theorem 3.8, and furthermore the modular
structure ensures that a simple 0-eigenvalue implies absence of non-zero eigenvalues, cf. Theorem 3.9. These results were derived from the underlying algebraic
structure of standard Liouvilleans, and may not have natural operator theoretic
proofs. It would be natural to ask if it is not possible to extract even more information from the underlying algebraic framework.
Problem 3.5. Can one exploit the underlying algebraic structure to infer more information on the point spectrum and pertaining eigenstates, than what is afforded
by Theorem 3.9. In particular, can one use algebraic arguments to conclude that
zero is at most a simple eigenvalue of Lβ ?
It is well known that establishing instability or outright absence of embedded
eigenvalues away from zero coupling, or some other explicitly solvable regime,
is a daunting task. It is for example not known if embedded (necessarily negative) eigenvalues of N -body Schrödinger operators are unstable under perturbations of pair-potentials. One can only show generic instability under perturbations
26
by external potentials cf. [2, 1]. In [17] a Fermi Golden Rule was established at
arbitrary coupling for the Hamiltonian, but to conclude instability one needs better control of eigenstates beyond the ground state (where Perron-Frobenius theory
applies). The case of perturbation around zero coupling is far better understood
[7, 8, 9, 13, 14, 18, 25, 40]. Hence, whether or not the kernel of the standard Liouvillean is generically one-dimensional beyond the weak-coupling regime is not a
question one is likely to answer using perturbation theory of embedded eigenvalues
only.
We stress that we consider Problem 3.5 to be the most important problem highlighted in these notes. The reason being that, due to Theorem 3.9, it reduces the
question of establishing return to equilibrium beyond the weak coupling regime to
positive commutator estimates and limiting absorption principles. Something we
see no fundamental obstacle to obtaining, although the picture is not yet entirely
clear. See Subsect. 4.5.
Finally, it would be natural, in the spirit of [11], to investigate the types of
ultraviolet and infrared behavior of the standard Liouvillean when ν = 1, which
is a solvable case. See also the discussion on ground states versus β-KMS states
when ν = 1 in the previous subsection, which indicates that the infrared type II
property, cf. [11], characterizes existence of β-KMS states.
Problem 3.6. Classify possible types of ultraviolet and infrared behavior of the
“van Hove Liouvillean”, i.e. when ν = 1.
4 Commutator Estimates
4.1 The Weak Coupling Regime
The weak coupling regime is very well understood. To explore it we replace G
by λG, where λ ∈ R is small in norm. In fact, obtaining positive commutator
estimates in this regime is an easy exercise. Indeed, choosing a to be generator of
radial translation (3.16) we get using (2.2)
1
H ′ = N − λφ(iaG) ≥ N − 4λ2 kaGk2 .
2
Choosing λ such that 4λ2 kaGk2 ≤ 1/4 yields
1
1
H ′ ≥ 1lH − 1lK ⊗ |0ih0|.
4
4
We can now prove
27
(4.26)
Corollary 4.1. Suppose (HG1). There exists λ0 > 0 such that for λ ∈ [−λ0 , λ0 ]
the pure point spectrum σpp (H) is finite and all eigenvalues have finite multiplicity.
(Here H is defined with G replaced by λG.)
Proof. Let λ0 > 0 be such that (4.26) is satisfied for |λ| ≤ λ0 . Assume towards
a contradiction that there exists an enumerable sequence ψn of normalized eigenstates. From (4.26) and Theorem 3.5 we find that
hψn , 1lK ⊗ |0ih0|ψn i ≥ 1.
This is a contradiction since ψn converges weakly to zero and 1lK ⊗ |0ih0| is a
compact operator. Recall that K was assumed finite dimensional.
d
Similarly for the Liouvillean where we can again choose ã = i dω
⊗ 1lL2 (S 2 ) to
be the generator of translations in the glued variable. Then
e L − 4λ2 ãG
e β 2 .
e′ = N
e L − λφ(iãG
eβ ) ≥ 1 N
L
β
2
e β k)−2 we arrive at
Hence choosing |λ| ≤ λ0 < (2kãG
Corollary 4.2. Suppose (LG2). There exists λ0 > 0 such that for λ ∈ [−λ0 , λ0 ]
the pure point spectrum σpp (Lβ ) is finite and all eigenvalues have finite multiplicity. (Here Lβ is defined with G replaced by λG.)
Proof. The proof is identical to the proof of Corollary 4.1, except we make use of
Theorem 3.6 instead of Theorem 3.5.
This theorem improves on a result of Merkli [40] due to the improvement in
the number bound Theorem 3.4. See also [19, 18, 20].
4.2 Conjugate Operators
Let χ ∈ C0∞ (R) satisfy χ(−ω) = χ(ω), 0 ≤ χ ≤ 1, χ(ω) = 1 for |ω| ≤ 1/2 and
χ(ω) = 0 for |ω| > 1.
Let µ > 0 be the constant used to define the class of couplings we can treat,
cf. (HGn) and (LGn). We use it to construct an auxiliary function d : (0, ∞) →
[1, ∞) as follows
d(ω) = χ(ω)ω −µ/4 + χ(ω/2) − χ(ω) + (1 − χ(ω/2))ω µ/4 .
We leave it to the reader to verify the following properties of d
(d1) (ω − 1)d′ (ω) ≥ 0.
28
(d2) limω→0+ d(ω) = limω→+∞ d(ω) = +∞.
(d3) ∃C > 0 s.t. |d′ (ω)| ≤ Cd(ω)/ω for all ω > 0.
We extend d to R\{0} be setting d(ω) = d(−ω) for ω < 0.
For a given
δ = (δ0 , δ∞ ) ∈ (0, 1] × [1, ∞) =: ∆0
(4.27)
we define a smooth positive function mδ : R → [1, ∞) by
mδ (ω) = d(δ0 )χ(ω/δ0 )
+ d(ω) χ(ω/(2δ∞ )) − χ(ω/δ0 )
+ d(δ∞ ) 1 − χ(ω/(2δ∞ )) .
Observe that mδ has bounded and compactly supported derivatives.
Our conjugate operator on the one-particle level for the Hamiltonian at zero
temperature is defined as the modified generator of radial translations
i
k
k
aδ =
mδ (k)
· ∇k + ∇k ·
mδ (k) .
2
|k|
|k|
Note that aδ a priori defined on C0∞ (R3 \{0}) is closable and its closure is a maximally symmetric operator. The conjugate operator is obtained through second
quantization
Aδ = 1lK ⊗ dΓ(aδ )
and is again a maximally symmetric operator closable on K ⊗ Γfin (C0∞ (R3 \{0})).
To get a conjugate operator for the Liouvillean we do the construction after
gluing and define the modified generator of translations
i
d
d
ãδ :=
mδ (ω)
+
mδ (ω) ⊗ 1lL2 (S 2 ) ,
2
dω dω
which is essentially self-adjoint on C0∞ (R) ⊗ C ∞ (S 2 ). We second quantize to
obtain
eδ := 1lK⊗K ⊗ dΓ(ãδ ),
A
which is essentially self-adjoint on CeL . Note that we have simplified the notation
eδ instead of the more cumbersome ãm and A
em , cf. (3.18)
a bit writing ãδ and A
δ
δ
and (3.19).
Note that as an identity on C0∞ (R\{0}) ⊗ C ∞ (S 2 ) we have T (aδ ⊗ 1lh − 1lh ⊗
aδ )T ∗ = ãδ and furthermore
e ′∞ U ∗ = L′∞ = H ′ ⊗ 1lH + 1lH ⊗ H c′ .
UL
The latter being an operator identity on D(N L ).
29
(4.28)
4.3 Estimates at Zero Temperature
Throughout this section we will for δ ′ ∈ ∆0 , cf. (4.27), use the notation
′
∆(δ′ ) := δ ∈ ∆0 δ0 ≤ δ0′ , δ∞ ≥ δ∞
.
We begin with a new high-energy estimate, which is particular to the case of finite dimensional small systems. It will not hold e.g. for (confined) atomic small
systems.
For δ ∈ ∆0 , we will write Nδ for dΓ(mδ ), the modified number operator appearing in H ′ = Nδ − φ(iaδ G). The reason for introducing the modified generator
of radial translation is that Nδ is large in the infrared and ultraviolet regimes, which
allows us to handle very soft and very hard photons.
′ ≥ 1, c > 0
Theorem 4.3. Suppose (HG1). Let e > 0 be given. There exists δ∞
′ )) we have
and E0 ∈ R such that for all δ ∈ ∆((1, δ∞
H ′ ≥ e1lH − c1l[H ≤ E0 ]
in the sense of forms on D(N ).
Proof. The first step we take is to estimate from below
1
H ′ ≥ Nδ − C1lH
2
(4.29)
exploiting the N 1/2 -boundedness of φ(iaδ G), cf. (2.2), and the inequality N ≤
Nδ . Here C is some positive number.
For R > 1 we perform a partition of unity in momentum space as follows. Let
1l[|k| < R]
R
R
F =
: h → L2 (B(0, R)) ⊕ L2 (B(0, R)c ) =: hR
< ⊕ h> (4.30)
1l[|k| ≥ R]
and observe that F R is unitary. We lift to F and get
R
R
R
Γ̌(F R ) = IΓ(F R ) : F → Γ(hR
< ) ⊗ Γ(h> ) =: F< ⊗ F> .
(4.31)
Put H0x = K ⊗ 1l + 1lK ⊗ Hph |F R ⊗ 1l + 1l ⊗ Hph |F R and abbreviate λmax =
<
>
30
e > λmax
max σ(K). We now compute for E
e
Nδ ≥ Nδ 1l[H0 > E]
n
o
e Γ̌(F R )
= Γ̌(F R )∗ 1lK ⊗ Nδ |F R ⊗ 1lF>R + 1lK⊗F<R ⊗ Nδ |F R 1l[H0x > E]
<
>
x
R
e + mδ (R)Γ̌(F R )∗ 1l
e
≥ Γ(1l[|k| < R])Nδ 1l[H0 > E]
R ⊗ P Ω 1l[H0 > E]Γ̌(F )
K⊗F<
e − λmax
E
x
R
e + mδ (R)Γ̌(R )∗ 1l
e
Γ(1l[|k| < R])1l[H0 > E]
R ⊗ P Ω 1l[H0 > E]Γ̌(F )
K⊗F<
R
(
)
e − λmax
E
e
≥ min
, mδ (R) 1l[H0 > E].
R
≥
We thus get
1
1
Nδ − C1lH ≥ min
2
2
(
)
e − λmax
E
e − C1lH .
, mδ (R) 1l[H0 > E]
R
(4.32)
To pass from H0 to H we estimate, recalling that Σ denotes the bottom of the
spectrum of H (2.4),
e ≤ (E
e + 1)(H0 + 1)−1
1l[H0 ≤ E]
1
e + 1)(H − Σ + 1)− 2
= (E
n
o
1
1
1
−1
2
2
× (H − Σ + 1) (H0 + 1) (H − Σ + 1) (H − Σ + 1)− 2
e + 1)C(H
e
≤ (E
− Σ + 1)−1
e
e
e + 1)C1
e l[H ≤ E] + (E + 1)C .
≤ (E
E −Σ+1
Combining with (4.29) and (4.32) we arrive at the bound
)
(
!
e E
e + 1)
e − λmax
e + 1)C
e
1
C(
E
(E
H ′ ≥ min
1−
, mδ (R)
1l[H ≤ E].
−C−
2
R
E−Σ+1
2
We are now ready to pick our constants. First choose R large enough such that
′ ≥ R such that for δ ∈ (0, 1) × (δ ′ , ∞)
d(R)/2 ≥ e + C + 1. Then choose δ∞
∞
e large enough such that (E
e−
we have mδ (R) = d(R). Subsequently we fix E
e (in that order) we get
λmax )/(2R) ≥ e + C + 1. With these choices of R, δ ′ and E
!
e + 1)C
e
e E
e + 1)
(E
C(
1l[H ≤ E].
H ′ ≥ (e + C + 1) 1 −
−C −
E−Σ+1
2
Finally we can take E0 large enough such that with E = E0 the right-hand side is
e E
e + 1)1l[H ≤ E0 ].
bounded from below by e1lH − 21 C(
31
For the purpose of the following we introduce the terminology that H satisfies
a Mourre estimate at E ∈ R:
Definition 4.4. We say that H satisfies a Mourre estimate at E ∈ R if there exists
δ ′ ∈ ∆0 such that: For all ǫ > 0 there exist C > 0, κ > 0 and a compact operator
K such that as a form on D(N )
H ′ ≥ (1 − ǫ)1l − C1l[|H − E| > κ] − K,
for any δ ∈ ∆(δ ′ ).
Theorem 4.3 implies that a Mourre estimate is satisfied at any E > E0 . Since
is bounded from below, we also get a Mourre estimate automatically satisfied
at any E < Σ. Note that the Mourre estimate obviously remains true if we replace
κ by any smaller positive κ′ .
H′
Lemma 4.5. Suppose (HG1). Let J ⊂ R be a compact set with a Mourre estimate
satisfied at all E ∈ J. Then there exists δ ′ ∈ ∆((1, 1)) such that: For any ǫ > 0,
there exist κ > 0, C > 0, such that for all E ∈ J and δ ∈ ∆(δ ′ ) we have
H ′ ≥ −ǫ1lH − C1l[|H − E| ≥ κ],
in the sense of forms on D(N ).
Remark 4.6. If a Mourre estimate holds at all E ∈ J with the same δ ′ , then this δ ′
can also be used for the uniform bound. This will be evident from the proof below.
♦
Proof. First note that by the virial theorem, the point spectrum in an open neighborhood of J is locally finite and eigenvalues in J have finite multiplicity.
We begin by verifying the estimate for a fixed E, for which the Mourre estimate
is satisfied. If E 6∈ σpp (H) we proceed as follows: First extract a Mourre estimate
with the given ǫ. Write for 0 < κ′ < κ the compact error as K = K1l[|H − E| ≤
κ′ ] + K1l[|H − E| > κ′ ]. Pick κ′ small enough such that kK1l[|H − E| ≤ κ′ ]k ≤
1/2. Then
H ′ ≥ −ǫ1l − (C + kKk)1l[|H − E| > κ′ ].
If on the other hand E ∈ σpp (H) we proceed differently. Write PE for the
finite rank orthogonal projection on the eigenspace associated with E. Abbreviate
32
P E = 1l − PE . Since Ran(PE ) ⊂ D(N 1/2 ) we can compute and estimate
H ′ = PE H ′ PE + 2Re{PE H ′ P E } + P E H ′ P E
= 2Re{PE H ′ P E } + P E H ′ P E
≥ (1 − ǫ/5)P E − C1l[|H − E| ≥ κ] − P E KP E + 2Re{PE H ′ P E }
≥ −ǫ/51l − C1l[|H − E| ≥ κ′ ] + 2Re{PE H ′ P E },
(4.33)
where we used Theorem 3.5 in the second equality and for the first inequality we
used the assumed to hold Mourre estimate (applied with ǫ replaced by ǫ/5). In the
last step we argued as above to get rid of the compact error by passing to a smaller
κ′ < κ.
As for the cross term (4.33) we write H ′ = Nδ − φ(iaδ G) as a form sum on
D(N 1/2 ). Recalling Theorem 3.1, we decompose for an r > 0 to be fixed later
PE H ′ P E = PE Nδ 1l[Nδ > r]P E + KP E ,
with K = PE Nδ 1l[Nδ ≤ r] − PE φ(iaδ G) being compact. Estimate first for σ > 0
2Re{PE Nδ 1l[Nδ > r]P E } = 2Re{PE Nδ 1l[Nδ > r]} − 2PE Nδ 1lNδ >r PE
≥ −σNδ − (2 + σ −1 )PE Nδ 1l[Nδ > r]PE .
fix σ small enough such that
8σkaδ Gk2 ≤ ǫ/5.
(4.34)
Fix now r large enough such that kPE Nδ 1l[Nδ > r]PE k ≤ ǫ/(5(2 + σ −1 )). We
then have
ǫ
2Re{PE Nδ 1l[Nδ > r]P E } ≥ −σNδ − 1lH .
5
′
Secondly we estimate for σ > 0
2Re{KP E } = 2Re{K1l[|H − E| < κ]P E } + 2Re{K1l[|H − E| > κ]}
≥ −(ǫ/5 + σ ′ )1l −
kKk2
1l[|H − E| > κ].
σ′
Here we chose κ > 0 small enough such that kK1l[|H − E| < κ]P E k ≤ ǫ/5.
Picking σ ′ = ǫ/5 yields
H′ ≥ −
4ǫ
1lH − σNδ − C1l[|H − E| > κ].
5
33
To get rid of the extra σNδ we estimate using (2.2) (with σ = 1/2) and (4.34)
4ǫ
1l − C1l[|H − E| > κ] + σNδ − 2σφ(iaδ G)
5
≥ −ǫ1l − C1l[|H − E| > κ].
(1 + 2σ)H ′ ≥ −
It remains to establish that one can choose δ ′ , κ and C such that the desired
bound holds for all E ∈ J and δ ∈ ∆(δ ′ ). We proceed by assuming, aiming for
a contradiction, that given δ ′n = (1/n, n), κn = 1/n and Cn = n, there exists an
energy En and δ n ∈ ∆(δ ′n ) such that the desired bound fails. By compactness of
J, we can assume that En converges to some E∞ . Let δ ′∞ , κ∞ and C∞ be the
constants just established to exist such that the bound holds true at E∞ for any
δ ∈ ∆(δ′∞ ). Picking n large enough such that
|E∞ − En | < κ∞ /2,
κn < κ∞ /2,
Cn ≥ C∞ ,
δ ′n ∈ ∆(δ ′∞ )
sets us up with a contradiction since with δ ∈ ∆(δ ′n ) ⊂ ∆(δ ′∞ ) we have
H ′ ≥ −ǫ1l − C∞ 1l[|H − E∞ | ≥ κ∞ ] ≥ (1 − ǫ)1l − Cn 1l[|H − En | ≥ κn ].
The following theorem, which appeared originally in [23, Thm. 7.12], states
that a Mourre estimate holds at any fixed E ∈ R. It holds also for confined small
system, not necessarily finite dimensional, but the proof simplifies slightly here.
Furthermore, since we do not need resolvents of H to control φ(aδ G) but can
do with resolvents of N , the version here in fact holds under slightly weaker IR
assumptions on G.
Another special feature of finite dimensional small systems is that we can
choose δ0′ uniformly in energy. Indeed, we pick δ0′ ∈ (0, 1] such that
d(δ0′ ) ≥ 2 sup kaδ Gk2 + 1.
(4.35)
δ∈∆0
With this choice we have for all δ ∈ ∆((δ0′ , 1)) and |ω| ≤ δ0 that
mδ (ω) ≥ mδ (δ0 ) = d(δ0 ) ≥ d(δ0′ ) ≥ 2 sup kaδ Gk2 + 1.
(4.36)
δ∈∆0
Theorem 4.7. Suppose (HG1). Let ǫ > 0 and E ∈ R. There exist κ > 0, C > 0
and K a compact and self-adjoint operator, such that the form estimate on D(N )
H ′ ≥ (1 − ǫ)1lH − C1l[|H − E| ≥ κ] − K
holds true for all δ ∈ ∆((δ0′ , 1)). Here δ0′ is chosen such that (4.35) is satisfied.
34
Proof. Fix E ∈ R and ǫ < 0. We only have something to prove if E ≥ Σ. The
proof goes by induction in energy and we assume the theorem holds true for all
E ′ ≤ E − δ0′ and ǫ′ > 0.
Write P = |0ih0| and P ⊥ = 1lF −P as projection operators on F or H (read as
e.g. 1lK ⊗ P ). In order to use geometric localization we need the extended Hilbert
space Hx = H ⊗ F and the extended Hamiltonian H x = H ⊗ 1lF + 1lH ⊗ Hph .
The extended commutator is
′
H x = H ′ ⊗ 1lF + 1lH ⊗ Nδ
as a self-adjoint operator on D(N x ), where N x = N ⊗ 1lF + 1lH ⊗ N .
Observe that if S : D(N 1/2 ) → Hx is bounded then for any σ > 0 we have
Re Γ̌(j R )∗ (1lH ⊗ P )S = Re Γ(j0R )S0
= Re 1l[|H − E| > 1]Γ(j0R )S0 (N + 1lH )−1/2 (N + 1lH )1/2 − K1
≥ −σ(N + 1lH ) − σ −1 kS0 (N + 1lH )−1/2 k2 1l[|H − E| > 1] − K1 .
Here S0 : D(N 1/2 ) → H ⊗ C is the first component of S and K1 is compact.
The observation above implies that we can pick R0 > 0 large enough such that
for R ≥ R0 we have for σ > 0 as a form on D(N )
′
H ′ ≥ Γ̌(j R )∗ H x Γ̌(j R ) − σ(N + 1lH )
′
≥ Γ̌(j R )∗ (1lH ⊗ P ⊥ )H x (1lH ⊗ P ⊥ )Γ̌(j R )
− σ(N + 1lH ) − C2 1l[|H − E| > 1] − K2 .
(4.37)
Here we employed the observation above with an (R, δ)-dependent operator S(R, δ) =
−(φ(iaδ G) ⊗ 1lF )Γ̌(j R ), which has kS0 (R, δ)(N + 1lH )−1/2 k bounded uniformly
in R ≥ R0 and δ ∈ ∆0 .
Fix σ such that
1 − 4ǫ
ǫ
ǫ
5
σ ≤ , 8σkaδ Gk2 ≤
and
≥ 1 − ǫ.
(4.38)
5
5
1 + 2σ
We now employ the momentum partition of unity from the proof of Theorem 4.3, cf. (4.30) and (4.30). Let F δ0 = (1l[|k| ≥ δ0 ], 1l[|k| < δ0 ]) and reδ0
δ0
is unitary. (For notational convenience be⊗ F<
call that Γ̌(F δ0 ) : F → F>
low, we have switched the order of the interior and exterior regions.) Abbreviate
δ0
δ0
b x = (1lH ⊗ Γ̌(F δ0 ))Hx = H ⊗ F>
H
⊗ F<
. Compute the intertwining relations
Γ̌(F δ0 )P ⊥ = 1lF δ0 ⊗ P<⊥ + P>⊥ ⊗ P< Γ̌(F δ0 )
(4.39)
>
′
′
b x 1lH ⊗ Γ̌(F δ0 )
(4.40)
1lH ⊗ Γ̌(F δ0 ) H x = H
b x 1lH ⊗ Γ̌(F δ0 ) .
1lH ⊗ Γ̌(F δ0 ) H x = H
(4.41)
35
δ0
,
Here P>/< denote the orthogonal projections onto the vacuum sectors inside F>/<
and
b x′ = H x′ ⊗ 1l δ0 + 1lH ⊗ 1l δ0 ⊗ N δ0 ,
H
>
F
F
δ|F
<
x′
H>
>
<
′
= H ⊗ 1lF δ0 + 1lH ⊗ 1lF δ0 ⊗ Nδ|F δ0 ,
>
>
>
b x = H ⊗ 1l δ0 ⊗ 1l δ0 + 1lH ⊗ H
H
δ ⊗ 1l δ0 + 1lH ⊗ 1l δ0 ⊗ H
δ .
F
F
ph|F 0
F
F
ph|F 0
>
<
>
<
>
<
Using that H ′ ≥ −2kaδ Gk2 , cf. (2.2), we estimate
x′
⊗ 1lF δ0 + 1lH ⊗ 1lF δ0 ⊗ Nδ|F δ0
1lH ⊗ 1lF δ0 ⊗ P<⊥ H>
<
<
>
>
2
⊥
≥ m(δ0 ) − 2kaδ Gk 1lH ⊗ 1lF δ0 ⊗ P<
(4.42)
x′
⊗ 1lF δ0 + 1lH ⊗ 1lF δ0 ⊗ Nδ|F δ0
1lH ⊗ P>⊥ ⊗ P< H>
<
>
<
⊥
⊥
x′
= (1lH ⊗ P> )H> (1lH ⊗ P> ) ⊗ P< .
(4.43)
>
and observe the identity
Using the intertwining relations (4.39) and (4.40), together with (4.42), (4.43)
and the choice of δ0′ , cf. (4.36), we get
′
(1lH ⊗ P ⊥ )H x (1lH ⊗ P ⊥ )
(4.44)
n
o
′
x
(1lH ⊗ P>⊥ ) ⊗ P< 1lH ⊗ Γ̌(F δ0 ) .
≥ 1lH ⊗ Γ̌(F δ0 )∗ 1lH ⊗ 1lF δ0 ⊗ P<⊥ + (1lH ⊗ P>⊥ )H>
>
To deal with the term in the brackets we note that
′
x
(1lH ⊗ P>⊥ ) ≥ 1lH ⊗ P>⊥ + H ′ ⊗ 1lF δ0
(1lH ⊗ P>⊥ )H>
(4.45)
>
and estimate using Lemma 4.5, with ǫ replaced by ǫ/5, and the induction assumption
)
(∞ Z
M ⊕
H ′ dk1 · · · dkℓ ⊗ P<
H ′ ⊗ 1lF δ0 ⊗ P< =
>
≥−
(
∞ Z
M
ℓ=1
ℓ=1
⊕
(R3 \B(δ0 ))ℓ
(R3 \B(δ0 ))ℓ
P
ǫ1lH + C1l[|H + ℓj=1 |kj | − E| > κ] dk1 · · · dkℓ
ǫ
b x − E| > κ]1lH ⊗ P ⊥ ⊗ P<
= − 1lH ⊗ P>⊥ ⊗ P< − C1l[|H
>
n
ǫ
b x − E| > κ].
≥ − 1lHbx − C1l[|H
5
36
)
⊗ P<
(4.46)
Here κ and C are coming from Lemma 4.5. Combining (4.44)–(4.46), cf. also
(4.39) and (4.41), we find
′
(1lH ⊗ P ⊥ )H x (1lH ⊗ P ⊥ )
o
n
ǫ
b x − E| > κ] 1lH ⊗ Γ̌(F δ0 )
≥ 1lH ⊗ Γ̌(F δ0 )∗ 1lH ⊗ 1lF δ0 ⊗ P<⊥ + 1lH ⊗ P>⊥ ⊗ P< − 1lHb x − C1l[|H
>
5
ǫ
⊥
x
= 1lH ⊗ P − 1lHx − C1l[|H − E| > κ].
5
Pick a non-negative f ∈ C0∞ (R) with supp(f ) ⊂ [−κ, κ] and f = 1 on
[−κ/2, κ/2]. We are now in a position to insert into (4.37) and estimate for some
R ≥ R0 , which we can now fix,
n
o
H ′ ≥ Γ̌(j R )∗ 1lH ⊗ P ⊥ − ǫ1lHx − C1l[|H x − E| > κ] Γ̌(j R )
− σ(N + 1lH ) − C2 1l[|H − E| > 1] − K2
o
n
ǫ
≥ Γ̌(j R )∗ (1 − )1lHx − Cf (H x − E) Γ̌(j R )
5
− σ(N + 1lH ) − C3 1l[|H − E| > 1] − K3
2ǫ
≥ (1 − )1lH − σ(N + 1lH ) − Cf (H − E) − C3 1l[|H − E| > 1] − K3
5
2ǫ
≥ (1 −
− σ)1lH − σN − C4 1l[|H − E| > κ/2] − K3 .
5
with C4 = C + C3 .
The proof is now completed as in the proof of Lemma 4.5 by the bound
4ǫ
)1lH − C4 1l[|H − E| > κ/2] − K3 ,
5
where we used the choice of σ, cf. (4.38). This concludes the proof.
(1 + 2σ)H ′ ≥ H ′ + σN − 4σkaδ Gk2 ≥ (1 −
Repeating the proof of Corollary 4.1 we arrive at
Corollary 4.8. Suppose (HG1). The operator H has a finite number of eigenvalues, all of finite multiplicity.
We denote by P the finite rank projection that projects onto the subspace consisting of eigenstates for H, and we write P = 1l − P .
Corollary 4.9. Suppose (HG1). Let ǫ > 0. There exists δ ′ ∈ ∆0 , κ > 0, C > 0,
such that the following two estimates holds for all δ ∈ ∆(δ ′ ) and E ∈ R
H ′ ≥ −ǫ1l − C1l[|H − E| ≥ κ],
′
H ≥ (1 − ǫ)1l − C(1l[|H − E| ≥ κ] + P ),
in the sense of forms on D(N ).
37
(4.47)
(4.48)
Proof. The estimate (4.47) is a direct consequence of Theorems 4.3 and 4.7 together with Lemma 4.5.
We proceed to the second bound (4.48). This bound is obviously true for E >
E0 + 1 (cf. Theorem 4.3) and for E < Σ − 1, so what remains is to prove the
estimate uniformly in E ∈ [Σ − 1, E0 + 1] =: J, which is a compact interval.
We first argue that the estimate is correct for fixed E ∈ J. Let ǫ > 0 and apply
Theorem 4.7 with ǫ replaced by ǫ/3. Write
K = P KP + P KP + 2Re{P KP } ≥ −kKkP + P KP + 2Re{P KP }
and estimate
KP = KP 1l[|H − E| ≥ κ] + KP 1l[|H − E| < κ],
where one can choose κ small enough such that kKP 1l[|H − E| < κ]k ≤ ǫ/3. For
the first term we estimate
2hψ, KP 1l[|H − E| ≥ κ]ψi ≤ 2kKψkk1l[|H − E| ≥ κ]ψk
1
≤ σkKk2 kψk2 + hψ, 1l[|H − E| ≥ κ]ψi.
σ
Choosing σ > 0 small enough we get
e l[|H − E| ≥ κ].
K ≥ − 32 ǫ1l − C1
This completes the argument that for a fixed E one can find κ and C such that the
commutator estimate (4.48) holds true.
Suppose the estimate (4.48) is not correct uniformly in E. That is, for any
κ > 0 and C > 0 there exists E ∈ J such that estimate fails to hold.
Put κn = 1/n and Cn = n. This gives a sequence En ∈ J, for which the
estimate (4.48) is false. We can assume due to compactness of J that En converges
to an energy E∞ ∈ J. Recalling that we have just verified that (4.48) holds for a
fixed E ∈ J, we get a κ∞ > 0 and C∞ > 0 such that (4.48) holds true at E∞ .
Pick n large enough such that 1/n < κ∞ /2, Cn > C∞ and |E∞ − En | < κ∞ /2.
Then
H ′ ≥ (1−ǫ)1l−C∞ (1l[|H−E| ≥ κ∞ ]+P ) ≥ (1−ǫ)1l−Cn (1l[|H−En | ≥ κn ]+P ),
contradicting the choice of En .
38
4.4 Estimates at Positive Temperature
eδ for dΓ(mδ ), which is the analogue of Nδ
In this subsection we use the notation N
e β ).
e ′ = 1lK⊗K ⊗ N
eδ − φ(iãδ G
from the previous subsection. We can then write L
β
′ >0
Theorem 4.10. Suppose (LG1). Let e > 0 be given. There exists E0 > 0, δ∞
L
and C > 0 such that the following form bound holds on D(N ) for all E ≥ E0
′ ))
and δ ∈ ∆((1, δ∞
L′β ≥ e1l − C1l[|Lβ | < E].
e β and N
e.
Proof. It suffices to prove the theorem, with Lβ and N L replaced by L
The proof is divided into two steps. First we consider the uncoupled glued Lioue 0 . The reader should not confuse the subscript 0 with infinite temperature
villean L
(zero inverse temperature). We proceed as in the proof of Theorem 4.3.
Observe that
e ′0 = 1lK⊗K ⊗ N
eδ .
L
For R > 1 we again perform a partition of unity in momentum space as follows.
Let
1l[|ω| < R]
R
j =
: h̃ → h̃< ⊕ h̃>
1l[|ω| ≥ R]
h̃< := L2 ((−R, R)) ⊗ L2 (S 2 ),
h̃> := L2 ((−∞, R] ∪ [R, ∞)) ⊗ L2 (S 2 ).
Put Fe< = Γ(h̃< ), Fe> = Γ(h̃> ) and
e x = LK ⊗ 1l e e + 1lK⊗K ⊗ dΓ(ω) e ⊗ 1l e + 1l
L
e< ⊗ dΓ(ω)|F
e> .
0
F< ⊗F>
|F<
F>
K⊗K⊗F
Abbreviate λmax = max σ(K) and λmin = min σ(K) . We now estimate for
e > 2λmax − λmin
E
eδ 1l[|L
e0| + N
e ≥ E]
e
eδ ≥ 1lK⊗K ⊗ N
1lK⊗K ⊗ N
n
o
eδ
eδ
ex| + N
e x ≥ E]
e Γ̌(j R )
= Γ̌(j R )∗ 1lK⊗K ⊗ N
⊗
1
l
+
1
l
⊗
N
1l[|L
e
e
0
e
e
F>
K⊗K⊗F<
|F
|F
<
>
e0| + N
e ≥ E]
e
eδ 1l[|L
≥ Γ(1l[|ω| < R])N
ex| + N
e x ≥ E]
e Γ̌(j R )
+ mδ (R)Γ̌(j R )∗ 1lK⊗K⊗Fe< ⊗ P Ω 1l[|L
0
e − λmax + λmin
E
e0| + N
e ≥ E]
e
Γ(1l[|ω| < R])1l[|L
R+1
ex| + N
e x ≥ E]
e Γ̌(j R )
+ mδ (R)Γ̌(j R )∗ 1l ⊗ P Ω 1l[|L
0
(
)
e − λmax + λmin
E
e0| + N
e ≥ E].
e
≥ min
, mδ (R) 1l[|L
R+1
≥
39
To conclude the proof we estimate
e0| + N
e < E]
e ≤ (E
e + 1)(|L
e0| + N
e + 1)−1
1l[|L
1
e + 1)(|L
e β | + 1)− 2
= (E
o
n
e0| + N
e + 1)−1 (|L
e β | + 1) 21 (|L
e β | + 1)− 21
e β | + 1) 21 (|L
× (|L
e
e + 1)1l[|L
e β | < E] + C E + 1 .
≤ C(E
E +1
Here we used Proposition 2.4 2 to conclude that
e0| + N
e + 1)−1 (|L
e β | + 1) 21 < ∞.
e β | + 1) 21 (|L
C = (|L
e and E in that order as in the proof of Theorem 4.3 to
We can now pick R, δ ′ , E
conclude the proof.
It is now an immediate consequence of Theorem 3.6 that
Corollary 4.11. Suppose (LG2). The set of eigenvalues σpp (Lβ ) is bounded.
From now on we assume at least (HG1) and fix δ ′ such that Corollary 4.9 holds
true. Recall that (LG1) implies (HG1).
Proposition 4.12. Suppose (HG1). Let ǫ > 0 be given. There exists κ > 0 and
C > 0 such that for all E ∈ R and δ ∈ ∆(δ ′ )
L′∞ ≥ (1 − ǫ)1lHL − C 1l[|L∞ − E| ≥ κ] + P ⊗ P c
in the sense of forms on D(N L ).
Remark 4.13. Note that at zero temperature we do not need Nelson’s commutator
theorem to build L∞ , nor do we have any singularities from ρβ to absorb. Hence
we can work under an (HG1) condition instead of an (LG1) condition.
Proof. The starting point is the identity
L′∞ = H ′ ⊗ 1l + 1l ⊗ H c′ .
Denote by P ∈ B(H) the projection onto the span of all eigenstates of the operator
H. This is a finite range projection and hence compact. Put P c = CP C to be the
eigen projection onto the span of the eigenstates of H c . We write P = 1l − P and
c
P = 1l − P c . We deal with H ′ ⊗ 1l only since bounds on 1l ⊗ H c′ can be obtained
by conjugation with EC, where E is the exchange map that sends ψ ⊗ ϕ to ϕ ⊗ ψ.
Here ψ, ϕ ∈ H.
40
We write
H ′ = P H ′ P + 2Re{P H ′ P } + P H ′ P ,
(4.49)
which makes sense as forms on D(N 1/2 ) since P maps into D(N 1/2 ) by Theorem 3.1. We estimate each term differently. For the first and last term we use
(4.47) and (4.48) from Corollary 4.9 (applied with ǫ/9 instead of ǫ) and find
ǫ
ǫ
P H ′ P ≥ − P − C1l[|H − λ − E| > κ] ≥ − 1lH − C1l[|H − λ − E| > κ]
9
9
ǫ
′
P H P ≥ 1 − P − C1l[|H − λ − E| > κ].
9
(4.50)
As for the cross term P H ′ P we proceed in a fashion similar to what was done in
the proof of Lemma 4.5. Write for an r > 0
P H ′ P = P Nδ 1l[Nδ > r]P + KP ,
with K = P Nδ 1l[Nδ ≤ r] − P φ(iaδ G) being compact. We can now fix first σ
small enough and subsequently r large enough such that
ǫ
2Re{P Nδ 1l[Nδ > r]P } ≥ −σNδ − 1lH
9
and
ǫ
8σkaδ Gk ≤ ,
9
1 − 8ǫ
9
> 1 − ǫ.
1 + 2σ
(4.51)
To deal with the term KP we note that we can choose κ small enough such that
2kKP 1l[|H − λ| < κ]k ≤ ǫ/18 uniformly in λ. Indeed, there exists Λ such that
2kKP 1l[|H| > Λ]k ≤ ǫ/18 and hence by a covering argument there exists κ > 0
such that 2kKP 1l[|H − λ| < κ]k ≤ ǫ/18 uniformly in λ ∈ R. We thus get for all
λ, E ∈ R.
2Re{KP } = 2Re{KP 1l[|H − λ − E| > κ]} + 2Re{KP 1l[|H − λ − E| < κ]}
C
ǫ
ǫ
≥ − 1lH − 1l[|H − λ − E| > κ] − 1lH .
18
ǫ
18
Inserting this together with (4.50) into (4.49) we arrive at the bound
H′ ≥ 1 −
3ǫ
ǫ
P − 1lH − σNδ − C1l[|H − λ − E| > κ].
9
9
(4.52)
From the spectral theorem in multiplication operator form, we get a measure
space (M, Σ, µ), a measurable real function f on M and a unitary map U : H →
L2 (M) such that U HU ∗ = Mf , multiplication by f . Put U c = U C such that
U c H c U c∗ = Mf as well. Here U c∗ = CU ∗ . The combined map U L = U ⊗
41
U c : HL → L2 (M × M) (with product σ-algebra and measure) now sets up the
∗
correspondence U L L∞ U L = Mf1 −f2 , where fj (q1 , q2 ) = f (qj ).
Then under the identification L2 (M × M) = L2 (M; L2 (M)) we get
Z ⊕
1l[|Mf − f (q) − E| > κ] dµ(q).
1l[|Mf1 −f2 − E| > κ] =
M
Hence we conclude from (4.52) the estimate
Z ⊕
L ′
L∗
U H ′ U ∗ dµ(q)
U H ⊗ 1lH U =
M
Z ⊕
3ǫ
ǫ
≥
1 − U P U ∗ − 1lL2 (M) − σU Nδ U ∗ − C1l[|Mf − f (q) − E| > κ] dµ(q)
9
9
M
ǫ
3ǫ
∗
L
1 − P ⊗ 1lH − 1lHL − σNδ ⊗ 1lH − C1l[|L∞ − E| > κ] U L
=U
9
9
in the sense of forms on U L D(N L ). Adding to the above a similar bound for
1lH ⊗ H c′ yields
ǫ 6ǫ
c
P ⊗ 1lH + 1lH ⊗ P − 1lHL
9
9
L
− σN − C 1l[|L∞ − E| > κ] + P ⊗ P c
7ǫ
≥ (1 − )1lHL − σNδL − (C + 2) 1l[|L∞ − E| > κ] + P ⊗ P c .
9
L′∞ ≥ 1 −
Here we used that
c
c
P ⊗ 1lH + 1lH ⊗ P = 21lHL − P ⊗ P − P ⊗ P c − 2P ⊗ P c ≥ 1lHL − 2P ⊗ P c .
We now complete the proof, cf. (4.51), by estimating
(1 + 2σ)L′∞ ≥ 1 −
8ǫ 1lHL − C 1l[|L∞ − E| > κ] + P ⊗ P c .
9
as at the end of the proofs of Lemma 4.5 and Theorem 4.7.
In order to perturb around zero temperature, we first need to control the differeβ − G
e∞ .
ence G
Lemma 4.14. Suppose (LGn). For any β0 > 0 there exists C > 0 such that for
all β ≥ β0 we have
eβ − G
e∞ ≤ Cβ − 21 .
G
(4.53)
If n ≥ 1 we have furthermore that for all δ ∈ ∆0
eβ − G
e ∞ ≤ Cβ − 21 .
ãδ G
42
(4.54)
Proof.
We begin with (4.53). For p
simplicity we only consider the contribution
p
e ∞ , the other term ρeβ G
e ∞,R being similar.
( 1 + ρeβ − 1)G
We split into the infrared and ultraviolet regimes and estimate first for |ω| ≤ 1:
q
q
2
2 e ∞ (ω, Θ)2 ≤ C
1 + ρeβ (ω) − 1 |ω|2n+2µ.
1 + ρeβ (ω) − 1 G
Hence we can bound the L2 -norm squared of the contribution by a multiple of
Z 1 q
Z 1
2 2n+2µ
−1
(1 + ω)ω 2n−1+2µ dω,
1 + ρeβ (ω) − 1 ω
dω ≤ β
0
0
e ∞ . The integral is finite
where we simply discarded the −1 term coming from G
e
for all n ≥ 0. In fact, the effect of subtracting G∞ sits in the ultraviolet part where
|ω| ≥ 1. Here we estimate the L2 -norm squared by
p
2
Z ∞ q
2 −1−2µ
1 + ρeβ (1) − 1
1 + ρeβ (ω) − 1 ω
.
dω ≤
2µ
1
p
p
Since 1 + ρeβ (1) − 1 = 1/(1 − e−β ) − 1 ∼ e−β/2 inn the limit of large β, we
get for a fixed β0 > 0 a constant C = C(β0 ) such that for all β > β0 we have
(4.53) satisfied.
To establish (4.54) we observe that
p
q
∂ 1 + ρeβ
e
e
e
1 + ρeβ − 1 ãδ G∞ + mδ G∞
.
ãδ Gβ =
∂ω
The first contribution can be estimate exactly as above, using that n ≥ 1, and yields
an O(β −1/2 ) term. For the second term we compute
p
∂ 1 + ρeβ
β q
= − ρeβ 1 + ρeβ .
∂ω
2
In the infrared regime this can be dealt with easily since β ρeβ ≤ 1/|ω| and the extra
e ∞ . Recall that we assume n ≥ 1. For the
inverse power of ω can be absorbed into G
ultraviolet regime we get exponential decay in β from ρeβ (1) and we are done.
eβ − G
e ∞ ) under a (LG2) condiWe remark that a similar bound holds for ã2δ (G
tion but we do not need this.
Theorem 4.15. Suppose (LG1). Let ǫ > 0 be given. There exists β0 > 0, κ > 0
and C > 0 such that for all E ∈ R, δ ∈ ∆(δ ′ ), and β ≥ β0
L′β ≥ (1 − ǫ)1l − C 1l[|Lβ − E| ≥ κ] + P ⊗ P c
in the sense of forms on D(N L ).
43
Proof. From (4.28), Proposition 4.12, applied with ǫ/4 instead of ǫ, and Lemma 4.14
e)
we get as a form bound on D(N
eβ − G
e∞ )
e′ = L
e ′ − φ iãδ (G
L
∞
β
C1 ǫ
e − C 1l[|L
e ∞ − E| > κ] + P∞ .
1lHeL − σ N
≥ 1− −
4 σβ
e∞.
Here P∞ = U (P ⊗ P c )U ∗ is the projection onto the eigenstates of L
∞
Pick a non-negative f ∈ C0 (R) with supp(f ) ⊂ [−κ, κ] and f = 1 on
[−κ/2, κ/2]. Let f˜ be an almost analytic extension of f . Write
Z
e ∞ − E) − f (L
e β − E) = 1
e ∞ − η)−1 − (L
e β − η)−1 dη.
f (L
∂¯f˜(η) (L
π C
p
e -bounded, cf. Corollary 2.6,
e is of class C 1 (L
e β ) with [N
e, L
e β ]◦ being N
Since N
p
e β − η)−1 preserves D( N
e ) and from
we conclude from [16, Lemma 3.3] that (L
[16, (3.9)] we in fact find that there exists n and C such that
e β − η)−1 (N
e + 1)− 12 ≤ C |Imη|−1 + |Imη|−n .
e + 1) 21 (L
( N
It follows that
f (L
e ∞ − E) − f (L
e β − E) (N
e + 1)− 12 Z
1
eβ − G
e ∞ (L
e β − η)−1 (N
e + 1)− 21 dη
≤
|∂¯f˜(η)| |Imη|−1 φ G
π C
eβ − G
e ∞ (N
e + 1)− 12 .
≤ C φ G
Appealing to Lemma 4.14 we thus get
e ∞ − E| > κ] ≤ 1l eL − f (L
e ∞ − E)
1l[|L
H
C2
σβ
e β − E| > κ/2] + σ N
e + C2 .
≤ 1l[|L
σβ
e β − E) + σ N
e+
≤ 1lHeL − f (L
Choose first σ > 0 small enough such that
12σ
eβ < ǫ
ãδ G
4
β≥1,δ∈∆0
sup
and
1 − 3ǫ/4
> 1 − ǫ,
1 + 3σ
and subsequently β0 ≥ 1 large enough such that
C1
C2
ǫ
+
< .
σβ0 σβ0
4
44
(4.55)
With these choices we arrive at the bound
e − C 1l[|L
e β − E| ≥ κ/2] + P∞ .
e ′ ≥ 1 − ǫ 1l e L − 2σ N
L
β
H
2
e ′ from
We conclude the proof by the usual argument, i.e. bounding (1 + 3σ)L
β
below, cf. (4.55) and the previous proof.
We conclude
Corollary 4.16. Suppose (LG2). There exists β0 > 0 such that for all β ≥ β0 , the
Liouvillean Lβ has finitely many eigenvalues, all of finite multiplicity.
We remark that in a (β, G)-regime where a positive commutator estimate holds
we can under the (LG2) condition conclude that eigenstates ψ of the standard Liouvillean Lβ satisfy that ψ ∈ D(N L ). This is a consequence of [16] and improves
the basic number bound Theorem 3.4, without imposing further conditions on G.
4.5 Open Problems III
The by far most central open question relevant for this section is whether or not one
can establish a positive commutator estimate for the standard Liouvillean for arbitrary inverse temperature β and coupling G. We have an unsubstantiated inkling
eδ .
that it should be possible to use A
Problem 4.1. Establish, for arbitrary β and G, a positive commutator estimate
e β , possibly making use of the
for the Jakšić-Pillet glued standard Liouvillean L
eδ .
conjugate operator A
We remark that we have not in this section made use of the modular conjugation
J, cf. (2.9), which takes Lβ to −Lβ . This may be an extra ingredient to make use
of.
While one can establish positive commutator estimates for the Hamiltonian
also for infinite dimensional small systems, cf. [23], the situation is fundamentally different for standard Liouvilleans. To see this consider as the small system a
one-dimensional harmonic oscillator. Here the uncoupled Liouvillean L0 will have
point spectrum (a multiple of) Z, with each eigenvalue having infinite multiplicity.
Hence, one should not expect a positive commutator estimate with compact error terms, barring some mechanism to lift the infinite degeneracy by other means.
However, in the dipole approximation this model is explicitly solvable [4, 5] and
Könenberg in his thesis managed to handle perturbations of the Harmonic oscillator
potential [36]. Note that one can construct a small system where the Hamiltonian
K has compact resolvent and L0 has point spectrum which is dense in R! See also
[19, 20] where an atomic small system is considered, and positive commutator
methods are applied in the weak coupling regime.
45
Problem 4.2. What can be said about the general structure of point spectrum without the assumption of small coupling or a finite dimensional small system. Are
positive commutator estimates useful at all?
We emphasize that all the proofs from Subsect. 4.4 make essential use of K
being finite dimensional.
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49